Second-order nonlinearities, optimization

The study of chiral materials with nonlinear optical properties might lead to new insights to design completely new materials for applications in the field of nonlinear optics and photonics. For example, we showed that chiral supramolecular organization can significantly enhance the second-order nonlinear optical response of materials and that magnetic contributions to the nonlinearity can further optimize the second-order nonlinearity. Again, a clear relationship between molecular structure, chirality, and nonlinearity is needed to fully exploit the properties of chiral materials in nonlinear optics. 

Applications of continuum solvation approaches to MCSCF wavefunctions have required a more developed formulation with respect to the HF or DFT level. Even for an isolated molecule, the optimization of MCSFCF wavefunctions represents a difficult computational problem, owing to the marked nonlinearity of the MCSCF energy with respect to the orbital and configurational variational parameters. Only with the introduction of second-order optimization methods and of the variational parameters expressed in an exponential form, has the calculation of MCSCF wavefunction became routine. Thus, the requirements of the development of a second-order optimization method has been mandatory for any successful extension of the MCSCF approach to continuum solvation methods. In 1988 Mikkelsen el ol. [10] pioneered the second-order MCSCF within a multipole continuum model approach in a spherical cavity. Aguilar et al. [11] proposed the first implementation of the MCSCF method for the DPCM solvation model in 1991, and their PCM-MCSCF method has been the basis of many extensions to more robust second-order MCSCF optimization algorithms [12],... 

Reviewed in Zyss J, Chemla DS (1987) Quadratic nonlinear optics and optimization of the second-order nonlinear optical response of molecular crystals. In Chemla DS, Zyss J (eds) Nonlinear and optical properties of organic molecules and crystals. Academic Press, Orlando, p 23... 

The development of highly active third-order nonlinear optical materials is important for all-optical signal processing. In contrast to second-order nonlinear optical molecular systems, there are few rational strategies for optimizing the third-order nonlinear optical response of molecular materials. Unlike second-order materials, there exist no molecular symmetry restrictions for the observation of a third-order nonlinear optical response. It is the instantaneous... 

J. Zyss and D.S. Chemla, Quadratic Nonlinear Optics and Optimization of Second-Order Nonlinear Optical Response of Molecular Crystals, Eds. D.S. Chemla and J. Zyss, Academic Press, Orlando, FL, 1987, Vol. 1, pp. 23-191. 

In recent years there has been a growing interest in the search for materials with large macroscopic second-order nonlinearities [20-22] because of their practical utility as frequency doublers, frequency converters and electro-optic modulators [23] by means of second-harmonic generation, parametric frequency conversion (or mixing) and the electro-optic (EO) effect. They are described by X (2w w, u)), 0, w), respectively. In order to optimize... 

In order to obtain a useful material possessing a large second order nonlinear susceptibility tensor % 2) one needs to use molecules with a large microscopic second order nonlinear hyperpolarizability tensor B organised in such a way that the resulting system has no centre of symmetry and an optimized constructive additivity of the molecular hyperpolarizabilities. In addition, the ordered structure thus obtained must not loose its nonlinear optical properties with time. The nonlinear optical (NLO) active moieties which have been synthesized so far are derived from the donor-rc system-acceptor molecular concept (Figure 1). 

V. P. Rao, K. Y. Wong, A. K.-Y. Jen, and R. M. Mininni, Optimization of second-order nonlinear optical properties of push-pull conjugated chromophores using heteroaromatics, Proc. SPIE 2025 156 (1993). 

Plaquet, A, Guillaume, M., Champagne, B., Castet, F., Ducasse, L., Pozzo, J.L., and Rodriguez, V. (2008) In silico optimization of merocyanine-spiropyran compounds as second-order nonlinear optical molecular switches. Rhys. Chem. Chem. Phys., 10, 6223-6232. 

From numerous tests involving optimization of nonlinear functions, methods that use derivatives have been demonstrated to be more efficient than those that do not. By replacing analytical derivatives with their finite difference substitutes, you can avoid having to code formulas for derivatives. Procedures that use second-order information are more accurate and require fewer iterations than those that use only first-order information(gradients), but keep in mind that usually the second-order information may be only approximate as it is based not on second derivatives themselves but their finite difference approximations. 

M. Gevers and G. Bastin. Analysis and Optimization of Systems, chapter A Stable Adaptive Observer for a Class of Nonlinear Second-order Systems, pages 143-155. Springer-Verlag, 1986. 

This chapter discusses the fundamentals of nonlinear optimization. Section 3.1 focuses on optimality conditions for unconstrained nonlinear optimization. Section 3.2 presents the first-order and second-order optimality conditions for constrained nonlinear optimization problems. 

Remark 1 The resulting optimization model is an MINLP problem. The objective function is linear for this illustrative example (note that it can be nonlinear in the general case) and does not involve any binary variables. Constraints (i), (v), and (vi) are linear in the continuous variables and the binary variables participate separably and linearly in (vi). Constraints (ii), (iii), and (iv) are nonlinear and take the form of bilinear equalities for (ii) and (iii), while (iv) can take any nonlinear form dictated by the reaction rates. If we have first-order reaction, then (iv) has bilinear terms. Trilinear terms will appear for second-order kinetics. Due to this type of nonlinear equality constraints, the feasible domain is nonconvex, and hence the solution of the above formulation will be regarded as a local optimum. 

Part 1, comprised of three chapters, focuses on the fundamentals of convex analysis and nonlinear optimization. Chapter 2 discusses the key elements of convex analysis (i.e., convex sets, convex and concave functions, and generalizations of convex and concave functions), which are very important in the study of nonlinear optimization problems. Chapter 3 presents the first and second order optimality conditions for unconstrained and constrained nonlinear optimization. Chapter 4 introduces the basics of duality theory (i.e., the primal problem, the perturbation function, and the dual problem) and presents the weak and strong duality theorem along with the duality gap. Part 1 outlines the basic notions of nonlinear optimization and prepares the reader for Part 2. 

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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