Nonlinear second-order effects

The mechanism of the third-order three-body induction interactions is somewhat more complicated. It can be shown that one can distinguish three principal categories. The first mechanism is simply the interaction of permanent moments on the monomer C with the moments induced on B by the nonlinear (second-order) effect of the electrostatic potential of the monomer A plus contributions obtained... 

Nonlinear second order optical properties such as second harmonic generation and the linear electrooptic effect arise from the first non-linear term in the constitutive relation for the polarization P(t) of a medium in an applied electric field E(t) = E cos ot. 

In Equation 6, n (a>.) is the intensity independent refractive index at frequency u).,.0 Tlie sum in Equation 5 is over all the sites (n) the bracket, < >, represents an orientational averaging over angles 0 and . Unlike for the second-order effect, this orientational average for the third-order coefficient is nonzero even for an isotropic medium because it is a fourth rank tensor. Therefore, the first step to enhance third order optical nonlinearities in organic bulk systems is to use molecular structures with large Y. For this reason, a sound theoretical understanding of microscopic nonlinearities is of paramount importance. 

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)... 

Before we examine how second- and third-order N LO effects are related to nonlinear polarization, we briefly examine an important symmetry restriction on second-order NLO properties. From Eq. (5), we can see that P(E) = P(0) + xmE + x E2 + x i)E3+... and P -E) = P(0) - x 1)E + xp)E2 - x Eh... we can also see from Fig. 11.1 that P(E) + P(-E) if%(2) + 0. In a centra symmetric material, P(E) is necessarily equal to P(-E) and, therefore, P(0), and other even-order terms must be zero. Therefore, for second-order effects to be observed in a molecule or material, the molecule or material must be non-centrosymmetric. However, no such requirement applies to odd-order processes, such as third-order effects [Fig. 11.1 shows P(E) = P(-E) for a material with only linear and cubic susceptibilities non-zero]. 

Thus, in both cases, the molecular unit can be tailored to meet a specific requirement. A second crucial step in engineering a molecular structure for nonlinear applications is to optimize the crystal structure. For second-order effects, a noncentrosymmetrical geometry is essential. Anisotropic features, such as parallel conjugated chains, are also useful for third-order effects. An important factor in the optimization process is to shape the material for a specific device so as to enhance the nonlinear efficiency of a given structure. A thin-film geometry is normally preferred because nonlinear interactions, linear filtering, and transmission functions can be integrated into one precise monolithic structure. 

Through cascade second-order effects, the second-order optical nonlinearities result on a third-order optical effect in a multistep or cascade process. This process is a due to the existence of microscopic electric fields that are generated by second-order nonlinear aligning of molecular dipoles. 

DFWM experiments are of particular interest in order to characterize centrosymmetric isotropic materials, where there is no competition with second-order effects and the time-response of the nonlinearity has to be evaluated. 

In this context, both stress and strain are defined considering very small deformations. Under these conditions, the properties of the material are strictly constant and second-order effects (nonlinearities) are negligible. For example, to effect large shear strains, it is necessary to apply stresses to the material other than the obvious shear stress. While nonlinear effects are important, they are beyond the scope of this book, and the reader is advised to consult more advanced texts. 

Wave mixing of two electric fields can give rise to second-order effects of nonlinear optics [4]. One of these is the harmonic generation that converts the fundamental wavelength of a laser into its half (see Section 12.2.2). But, if electric fields at different frequencies are used, the response of a medium with sufficient second-order dielectric susceptibility can be frequency shifted to the sum and the difference of the two laser frequencies [4]. In particular, sum frequency generation (SFG) is often used to study surfaces and has found applications to examine catalytic combustion [9,36]... 

Note that the term accounting for effective transport in the axial direction has been neglected in this model, for the reasons given already in Sec. 11.6. This system of nonlinear second order partial differential equations was integrated by Froment using a Crank-Nicolson procedure [76,77], to simulate a multitubular fixed bed reactor for a reaction involving yield problems. 

Cascading. In most cases, the distinction between second- and third-order nonlinearities is evident from the different phenomena each produce. That distinction blurs, however, when one considers the cascading of second-order effects to produce third-order nonlinear phenomena (51). In a cascaded process, the nonlinear optical field generated as a second-order response at one place combines anew with the incident field in a subsequent second-order process. Figure 2 shows a schematic of this effect at the molecular level where second-order effects in noncentrosymmetric molecules combine to yield a third-order response that may be difficult to separate from a pure third-order process. This form of cascading is complicated by the near-field relationships that appear in the interaction between molecules, but analysis of cascaded phenomena is of interest, because it provides a way to explore local fields and the correlations between orientations of dipoles in a centros5nnmetric material (52). 

However, as more individual op-amps are combined, the resulting composite op-amp becomes less forgiving. Circuit layout, lead dress, and second-order effects become much more important. The diode D in Fig. 7.121 is included to prevent a large unstable mode of operation due to op-amp nonlinearities that may be excited when power is first applied or when large noise spikes exist (Budak, Wullink, and Geiger, 1980). Normally, the diode is off, and it is not included in the analysis. 

When a polymer is subject to an intense sinusoidal electric field such as that due to an intense laser pulse, Fourier analysis of the polarization response can be shown to contain not only terms in the original frequency co, but also terms in 2(0 and 3nonlinear response depends on the square of the intensity of the incident beam for 2co, and the third power for 3 . For the second-order effects, the system must have some asymmetry, as discussed previously. For poling, this means both high voltage and a chemical organization that will retain the resulting polarization for extended periods of time. Polymeric systems investigated have been of three basic types ... 

Two applications have been earlier identified where LB films might have great potential, namely nonlinear optical devices based on a second-order effect, and sensors. The research on these topics is quite advanced... 

Nonlinear optical devices based on second-order effects... 

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