Second-order material coefficients

Equations of this kind are called linear constitutive equations. The derivatives of the dependent variables with respect to the independent ones represent material coefficients. Recalling Eqs. (4.8), (4.9), and (4.10) they may also be written as partial second derivatives of the appropriate thermodynamic potential. Because of this second-derivative properly these coefficients are called second-order material coefficients. 

Table 4.3 Interrelations between second order material coefficients...

Typical values of the second-order nonlinear coefficient d for dielectric crystals, semiconductors, and organic materials used in photonics applications lie in the range d = 10 24 to 10 - (mks units, As/V2). Typical values of the third-order nonlinear coefficient x(3> for glasses, crystals, semiconductors, semiconductor-doped glasses, and organic materials of interest in photonics are x -3 = 10 34 to 10 29 (mks units). 

Meier et al. [98] have shown that DAST crystal is also a very interesting material for phase-matched parametric oscillation around the telecommunication wavelength at A = 1318 and 1542 nm. Their results (as shown in Table 9) indicate that the second-order NLO coefficient dill = lOlOpm/V at 1318nm for DAST crystal. 

We have heretofore had ample discussion of linear optical properties of ZnO and related materials. In this section, the nonlinear processes in ZnO are discussed, a topic that has been investigated in some detail. The research on nonlinear optical properties of semiconductors is motivated by electro-optic devices that can be used in telecommunications and optical computing as efficient harmonic generators, optical mixers, and tunable parametric oscillators, among others. The nonlinear optical properties such as second harmonic generation (SHG), that is, (2(0i, 2(02), and the sum frequency generation (SFG), that is, (materials characterization, particularly surfaces, because the second-order susceptibility coefficient is very sensitive to the change in symmetry (178,179). The crystal should be... 

Quantitative XRF analysis has developed from specific to universal methods. At the time of poor computational facilities, methods were limited to the determination of few elements in well-defined concentration ranges by statistical treatment of experimental data from reference material (linear or second order curves), or by compensation methods (dilution, internal standards, etc.). Later, semi-empirical influence coefficient methods were introduced. Universality came about by the development of fundamental parameter approaches for the correction of total matrix effects... 

Tg can be determined by studying the temperature dependence of a number of physical properties such as specific volume, refractive index, specific heat, etc. First-order transitions, such as the melting of crystals, give rise to an abrupt change or discontinuity in these properties. However, when a polymeric material undergoes a second-order transition, it is not the primary property (the volume), but its first derivative with respect to temperature, (the coefficient of expansion), which becomes discontinuous. This difference between a first and second-order transition is illustrated in Figure 10. 

Below T0 the material is in the glassy state. Compared with the crystal the glass shows a larger specific volume and heat content, but both quantities have a smaller temperature coefficient than in the melt (< ). The transition from melt to glass is often called a transition of the second order (2, 3) since it is not accompanied by finite changes of volume and enthalpy, but only by changes of their temperature coefficients. 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution... 

In this paper, an overview of the origin of second-order nonlinear optical processes in molecular and thin film materials is presented. The tutorial begins with a discussion of the basic physical description of second-order nonlinear optical processes. Simple models are used to describe molecular responses and propagation characteristics of polarization and field components. A brief discussion of quantum mechanical approaches is followed by a discussion of the 2-level model and some structure property relationships are illustrated. The relationships between microscopic and macroscopic nonlinearities in crystals, polymers, and molecular assemblies are discussed. Finally, several of the more common experimental methods for determining nonlinear optical coefficients are reviewed. 

As mentioned above, the powder SHG method is a useful technique for the screening of second-order nonlinear materials. However, because of the sensitivity of the SHG coefficients of crystalline materials to the orientational aspects of the molecular packing and because the measurement is performed on an essentially random distribution of microcrystalline particles, the powder SHG method is not generally useful for obtaining information about molecular hyperpolarizabilities. 

Two of the most important nonlinear optical (NLO) processess, electro-optic switching and second harmonic generation, are second order effects. As such, they occur in materials consisting of noncentrosymmetrically arranged molecular subunits whose polarizability contains a second order dependence on electric fields. Excluding the special cases of noncentrosymmetric but nonpolar crystals, which would be nearly impossible to design from first principles, the rational fabrication of an optimal material would result from the simultaneous maximization of the molecular second order coefficients (first hyperpolarizabilities, p) and the polar order parameters of the assembly of subunits. (1)... 

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