Macroscopic-microscopic nonlinearity

In the weak coupling limit, as is the case for most molecular systems, each molecule can be treated as an independent source of nonrlinear optical effects. Then the macroscopic susceptibilities X are derived from the microscopic nonlinearities 3 and Y by simple orientationally-averaged site sums using appropriate local field correction factors which relate the applied field to the local field at the molecular site. Therefore (1,3)... 

The Problem of Translating Microscopic to Macroscopic Optical Nonlinearity... 

This result implies that the energy equipartition relationship of Eq. (2.S) applies as well as the general definitions of Chapter I. Note that for Af m the variable turns out to be coupled weakly to the thermal bath. This condition generates that time-scale separation which is indispensable for recovering an exponential time decay. To recover the standard Brownian motion we have therefore to assiune that the Brownian particle be given a macroscopic size. In the linear case, when M = w we have no chance of recovering the properties of the standard Brownian motion. In the next two sections we shall show that microscopic nonlinearity, on the contrary, may allow that the Markov characters of the standard Brownian motion be recovered with increasing temperature. 

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in terms of an ensemble average over the microscopic nonlinear polarizability tensors of the individual constituents. 

Harper, A.W., S. Sun, L.R. Dalton, S.M. Garner, A. Chen, S. Kalluri, W.H. Steier, and B.H. Robinson. 1998. Translating microscopic optical nonhnearity to macroscopic optical nonlinearity The role of chromophore-chromophore electrostatic interactions. / Opt Soc Amer B 15 329-337. 

Translating microscopic optical nonlinearity to macroscopic optical nonlinearity. 

Actually, it has been shown that the highest second order susceptibility coefficients (macroscopic nonlinearity), for a given chromophore (microscopic nonlinearity), can be reached in the (polar) crystal classes 1, 2, m and mm2, while other polar crystal classes are less favourable [21]. 

The macroscopic second-order NLO effects are not only dependent on the value of molecular hyperpolarizability, but are also dependent on the orientations of the molecules in the unit cells. The relation between microscopic and macroscopic second-order NLO effects have been studied by Zyss et al. [37]. When there are no significant intermolecular effects, the lowest-order macroscopic optical nonlinearity can be expressed as the ten-... 

Flaving now developed some of the basic notions for the macroscopic theory of nonlinear optics, we would like to discuss how the microscopic treatment of the nonlinear response of a material is handled. Wliile the classical nonlinear... 

Further subclassification of nonlinear optical materials can be explained by the foUowiag two equations of microscopic, ie, atomic or molecular, polarization,, and macroscopic polarization, P, as power series ia the appHed electric field, E (disregarding quadmpolar terms which are unimportant for device appHcations) ... 

The term collectivism has sometimes been used to distinguish this AL philosophy from the more traditional top down and bottom up philosophies. Collectivism embodies the belief that in order to properly understand complex systems, such systems must be viewed as coherent wholes whose open-ended evolution is continuously fueled by nonlinear feedback between their macroscopic states and microscopic constituents. It is neither completely reductionist (which seeks only to decompose a system into its primitive components), nor completely synthesist (which seeks to synthesize the system out of its constituent parts but neglects the feedback between emerging levels). 

Nonlinear processes being basically of Intramolecular nature, corresponding terms In the macroscopic and microscopic dipoles expansions can be related by the following tensorlal summation (given here for SHG coefficients), following an oriented gas description ... 

However, its was found possible to infer all four microscopic tensor coefficients from macroscopic crystalline values and this impossibility could be related to the molecular unit anisotropy. It can be shown that the molecular unit anisotropy imposes structural relations between coefficients of macroscopic nonlinearities, in addition to the usual relations resulting from crystal symmetry. Such additional relations appear for crystal point group 2,ra and 3. For the monoclinic point group 2, this relation has been tested in the case of MAP crystals, and excellent agreement has been found, triten taking into account crystal structure data (24), and nonlinear optical measurements on single crystal (19). This approach has been extended to the electrooptic tensor (4) and should lead to similar relations, trtten the electrooptic effect is primarily of electronic origin. 

Unlike in bulk nonlinear spectroscopy experiments, the signal in nonlinear microscopy is generated within a volume that is on the order of an optical wavelength. The axial extent of this volume is often referred to as the interaction length, which denotes the length within which the incident fields interact to produce a nonlinear polarization in the material. Such microscopic interaction lengths yield signal interference profiles that can differ markedly from those observed in macroscopic spectroscopy. 

Finally, the combination of dendrimers and organometallic entities as fundamental building blocks affords an opportunity to construct an infinite variety of organometallic starburst polymeric superstructures of nanoscopic, microscopic, and even macroscopic dimensions. These may represent a promising class of organometallic materials due to their specific properties, and potential applications as magnetic ceramic precursors, nonlinear optical materials, and liquid crystal devices in nanoscale technology. 

In this book, we intend to bring forth some recently understood nonlinear features of electro-diffusion and thus show that the aforementioned reputation is unjust. Thus we hope to provide an interdisciplinary qualitative supplement to a few, by and large numerically oriented, texts on the mathematical modelling of semiconductor devices that have appeared in recent years [14]—[18]. The reader possibly will recognize distant macroscopic relatives of the above-mentioned microscopic ionic effects in some of the effects discussed here. 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

The basic remark is that linearity of the macroscopic law is not at all the same as linearity of the microscopic equations of motion. In most substances Ohm s law is valid up to a fairly strong field but if one visualizes the motion of an individual electron and the effect of an external field E on it, it becomes clear that microscopic linearity is restricted to only extremely small field strengths.23 Macroscopic linearity, therefore, is not due to microscopic linearity, but to a cancellation of nonlinear terms when averaging over all particles. It follows that the nonlinear terms proportional to E2, E3,... in the macroscopic equation do not correspond respectively to the terms proportional to E2, E3,... in the microscopic equations, but rather constitute a net effect after averaging all terms in the microscopic motion. This is exactly what the Master Equation approach purports to do. For this reason, I have more faith in the results obtained by means of the Master Equation than in the paradoxical result of the microscopic approach. 

Of course, the macroscopic equations cannot actually be derived from the microscopic ones. In practice they are pieced together from general principles and experience. The stochastic mesoscopic description must be obtained in the same way. This semi-phenomenological approach is remarkably successful in the range where the macroscopic equations are linear, see chapter VIII. In the nonlinear case, however, difficulties appear, which can only be resolved by the improved, but still mesoscopic, method of chapter X. 

A. Microscopic and Macroscopic Third-Order Optical Nonlinearities... 

As has been discussed in part I of this review,1 the microscopic hyperpolarizabilities of the ith order have their corresponding quantities at the macroscopic level in the form of nonlinear susceptibilities The macroscopic polarization is then given by... 

Let us now embed the renormalization group, Constructed in Chap. 8, iftto this general framework. As mentioned above, relation (8.5) shows that the RG we are searching for must be a nonlinear representation of the group of dilatations in the space of parameters. , n,/ e). These are the microscopic parameters of the model, and the representation shall leave macroscopic observables invariant. Furthermore we want the representation to show a nontrivial fixed point. In Sect. 8.2 we have constructed such a representation based on first order perturbation theory. The invariance constraint is obeyed within deviations of order 1+e 2, no = n(A = 1). Equations (8.38), (8.42) give the parameter flow under this nonlinear representation in the standard form (10.28),... 

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. 

Crystals. In the discussion of equation 11, it was pointed out that the macroscopic nonlinear coefficients could be related to the microscopic ones in a relatively straightforward manner. For the hypothetical crystal shown in Figure 5, the relationship is (5)... 

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