Nonlinear materials

This chapter describes the physical and chemical basis for the development of nlo polymers. To achieve a fast response, the nlo properties must derive from electronic excitation of active molecular species. Progress towards an understanding of structure-property relationships for these has resulted from the development of tractable quantum mechanical models. These have resulted in general rules helpful in the development of synthetic routes to molecules with large nlo coefficients. Additional constraints come into play if a technologically useful material is to be obtained. These can be further refined in the context of the device structure relevant to particular end applications. These considerations are discussed in the following section. The development of polymers with particular nlo properties is presented in the subsequent sections. 

As with all physical properties, the response of media to radiation is linear only for weak radiation fields. Because of this, it is often claimed that the development of the laser led to the discover of nonlinear optical phenomena, since it was first optical source intense enough to produce detectable nonlinear effects. This is not true, since a number of nonlinear effects can be produced by external electrical fields, e.g. the Pockels and Kerr effects. These effects were known long before lasers were developed. Lasers simply rendered other nonlinear optical effects more easily observable in the laboratory. 

Here P is the response of the medium to external radiation field components Ej, E, E/, and etc. are the first, second, third and 

In polymeric materials the degree of ordering is in general lower for example, polar order may be induced in a certain axial direction. 

However, the susceptibility tensors are simpler in this case owing to symmetry considerations [12]. The special case of perfectly orientated polymer chains has the simple property that the internal and external fields along the chain axis are equal [13]. 

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As discussed above, the nonlinear material response, P f) is the most connnonly encountered nonlinear tenn since vanishes in an isotropic medium. Because of the special importance of P we will discuss it in some detail. We will now focus on a few examples ofP spectroscopy where just one or two of the 48 double-sided Feymnan diagrams are important, and will stress the dynamical interpretation of the signal. A pictorial interpretation of all the different resonant diagrams in temis of wavepacket dynamics is given in [41]. 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

Wallace [15], [16] gives details on effects of nonlinear material behavior and compression-induced anisotropy in initially isotropic materials for weak shocks, and Johnson et ai. [17] give results for infinitesimal compression of initially anisotropic single crystals, but the forms of the equations are the same as for (7.10)-(7.11). From these results it is easy to see where the micromechanical effects of rate-dependent plastic flow are included in the analysis the micromechanics (through the mesoscale variables and n) is contained in the term y, as given by (7.1). 

Let s address the issue of nonlinear material behavior, i.e., nonlinear stress-strain behavior. Where does this nonlinear material behavior come from Generally, any of the matrix-dominated properties will exhibit some degree of material nonlinearity because a matrix material is generally a plastic material, such as a resin or even a metal in a metal-matrix composite. For example, in a boron-aluminum composite material, recognize that the aluminum matrix is a metal with an inherently nonlinear stress-strain curve. Thus, the matrix-dominated properties, 3 and Gj2i generally have some level of nonlinear stress-strain curve. 

It has been shown that the thermodynamic foundations of plasticity may be considered within the framework of the continuum mechanics of materials with memory. A nonlinear material with memory is defined by a system of constitutive equations in which some state functions such as the stress tension or the internal energy, the heat flux, etc., are determined as functionals of a function which represents the time history of the local configuration of a material particle. 

As a result of the unique molecular electronic properties and geometrical structure, liquid crystals also tend to be optically nonlinear materials meaning... 

Figure 10.6 gives a schemahc view of the test setup of the strain measurement. The boundary conditions in this stretched him method are modeled by hnite element analysis with nonlinear material properties. 

Table I lists a variety of organic nonlinear materials which have appeared in the literature their relative powder efficiencies, absorption cutoffs and /3 values (if available) are also provided. These materials are "typical" only in that they represent results from the few classes of organic compounds investigated to date, yet they are instructive in that one learns which molecular properties may be important. A few caveats are in order to avoid misinterpretation of the data in Table I. Except for compound 10 (19) all the powder efficiency and cutoff data are from our own measurements. Powder measurements were performed on ungraded samples using the Nd YAG output at 1.06/t as fundamental since powder efficiency is a function of particle size distribution and a variety of other factors (3) these values are only semiquantitative. The cutoff values are the wavelengths for which 10-4M solutions in ethanol (unless otherwise indicated) have no absorbance. The cutoff values will be similar to those found in crystal state except where intermolecular charge transfer is important in the crystal or the molecule is solvatochromic, this latter effect being quite common for cyanine dyes such as...

The second step of the nonlinear material optimization process will be detailed in the cases of three efficient nonlinear materials recently developed at CNET POM, MAP and NPP. For all these materials specific features were implemented at the molecular level so as to act on the crystalline structure in the required direction (i.e. away from centrosymmetry and toward an "optimal" crystalline structure which has been theoretically defined O, 4) and actually almost reached in the case of NPP. 

Although a great deal of data analysis is needed to obtain a value of the optical nonlinearity from this measurement, we can estimate the order of magnitude of the value. The estimate comes from a comparison of this data with that taken on a similar experiment using Si as the nonlinear material. (15) The two experiments used approximately the same laser power and beam geometry, and the linear reflectivity curves were similar in shape and size (the minimum value of reflectivity). Neglecting differences... 

Equality constraints. The equality constraints (30 in all) are the linear and nonlinear material and energy balances and the phase relations. 

Nonlinear ion traps, 15 662 Nonlinear materials, 14 680 Nonlinear optical materials, 17 442-460 advantage of, 17 448 classification of, 17 443—444 economic aspects of, 17 457-458 photorefractive materials, 17 457 second-order, 17 444r-453 third-order, 17 453—457 Nonlinear optical phenomena, 17 443 Nonlinear optics... 

Selection of the material model is another important factor to be considered. Some programs allow the user to specify plastic moment-rotation curves for beam elements. However, the more rigorous and most widely available method of defining nonlinear material properties is to specify the stress versus strain data, Plastic behavior is approximated at the section level in the former method whereas, the latter method tracks plastic behavior at the individual integration points (fibers) through the thickness of the member. Each method has its advantages and disadvantages. 

The plastic hinge nonlinear material model is easier to use but usually can not consider axial load effects. Plastic hinge locations must usually be predetermined and are usually limited to the ends of the member. Analysis results which include displacements and plastic hinge rotations which are directly comparable against acceptance criteria. ... 

The more rigorous stress/strain nonlinear material model, oflen referred to as the plastic zone method, is theoretically capable of handling any general cross section Both isotropic and kinematic hardening rules are usually available. This method is... 

Figure 38 SchSinblk Pf-tfepresentation of the optically MdOdecPt-pblarization in (a) centrosymmetric non-linear materials (b) non-centrosymmetric nonlinear materials...

The interest in efficient optical frequency doubling has stimulated a search for new nonlinear materials. Kurtz 316) has reported a systematic approach for finding nonlinear crystalline solids, based on the use of the anharmonic oscillator model in conjunction with Miller s rule to estimate the SHG and electro optic coefficients of a material. This empirical rule states that the ratio of the nonlinear optical susceptibility to the product of the linear susceptibilities is a parameter which is nearly constant for a wide variety of inorganic solids. Using this empirical fact, one can arrive at an expression for the nonlinear coefficients that involves only the linear susceptibilities and known material constants. 

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