Macroscopic nonlinear coefficient

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)... 

For symmetry reasons, the first macroscopic nonlinear coefficient is zero in unordered polymer materials. On the other hand, azo-dye polymers can exhibit very large values, which is interesting for applications in optical limiting and optical switching devices. We will consider the relationship between microscopic and macroscopic third-order susceptibilities. The most general equation for this relationship can be written as ... 

The first theoretical attempts in the field of time-resolved X-ray diffraction were entirely empirical. More precise theoretical work appeared only in the late 1990s and is due to Wilson et al. [13-16]. However, this theoretical work still remained preliminary. A really satisfactory approach must be statistical. In fact, macroscopic transport coefficients like diffusion constant or chemical rate constant break down at ultrashort time scales. Even the notion of a molecule becomes ambiguous at which interatomic distance can the atoms A and B of a molecule A-B be considered to be free Another element of consideration is that the electric field of the laser pump is strong, and that its interaction with matter is nonlinear. What is needed is thus a statistical theory reminiscent of those from time-resolved optical spectroscopy. A theory of this sort was elaborated by Bratos and co-workers and was published over the last few years [17-19]. 

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. 

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. 

In the literature however, other related parameters, besides x are often used to describe the macroscopic second-order NLO properties of materials. The SHG nonlinear coefficient d and the linear electro-optic coefficient r are the parameters commonly used for second-harmonic generation and the Pockels effect respectively [3, 5]. They are related to x according to Eqs. (4) and (5). 

Many organic molecules have been shown to exhibit large nonlinear optical coefficients [15]. Especially interesting for second-order nonlinear optics are molecules with conjugated double bonds and with an electron donor and acceptor attached to opposite ends. The donor-acceptor pair leads to an asymmetry in the molecule that is required for second-order nonlinear effects and introduces a permanent dipole moment ju,. However, arranging these molecules in material systems with large macroscopic nonlinearities is not trivial. As opposed to the case of third-order... 

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 nonlinear response of these materials has been the subject of theoretical investigations, based on the oriented gas model. In this model it is assumed that (i) chromophores don t interact with the matrix and with each other (ii) they have cylindrical shape, are free to rotate under the influence of the poling field and have only one significant beta component, i.e. that along the molecular axis Pxxx, (iii) the permanent dipole moment of the molecule is oriented along the z axis. On this basis, a relation between the macroscopic nonlinearity of the material (i.e. r33 or r/33 coefficient) and the molecular properties of the chromophore can be deduced [2]... 

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 ... 

This phenomenological identification of A and B has been utilized by Einstein and others with great success (section 3), but only for linear Fokker-Planck equations. If the macroscopic law is nonlinear a difficulty arises, first pointed out by D.K.C. MacDonald. The flaw in the argument lies in the identification of the coefficient A y) with the macroscopic law. The two may well differ by a term of the same order as the fluctuations once one neglects the fluctuations such a term is invisible anyway. The consequence was that different authors obtained different, but equally plausible expressions for noise in nonlinear systems. This difficulty led to the more fundamental approach in chapter X. 

Conclusion. For internal noise one cannot just postulate a nonlinear Langevin equation or a Fokker-Planck equation and then hope to determine its coefficients from macroscopic data. ) The more fundamental approach of the next chapter is indispensable. ... 

Comparison with (X.2.16) shows the following drastic differences. The dominant term of (X.2.16) is absent and therefore no equation for the macroscopic part of X can be extracted. In other words, on the macroscopic scale the system does not evolve in one direction rather than the other. The remaining evolution of P is merely the net outcome of the fluctuations. Accordingly the time scale of the change is a factor slower than in the preceding case, compare (X.2.14). Since P is not sharply peaked the coefficients a(x) cannot be expanded around some central value but they remain as nonlinear functions in the equation. The first line of (1.4) contains the main terms and is called the diffusion approximation... 

The cell sizes are expected to exceed any molecular (atomic) scale so that a number of particles therein are large, Ni(f) 1. The transition probabilities within cells are defined by reaction rates entering (2.1.2), whereas the hopping probabilities between close cells could easily be expressed through diffusion coefficients. This approach was successfully applied to the nonlinear systems characterized by a loss of stability of macroscopic structures and the very important effect of a qualitative change of fluctuation dispersion as the fluctuation length increases has also been observed [16, 27]. In particular cases the correlation length could be the introduced. The fluctuations in... 

The general solutions of the fundamental systems of nonlinear equations [Eq. (2)] will be of the type wherein the state variables are dependent both on time and space, which will manifest in the form of wave propagation. Coupling between several parts of the system will be transmitted through the generalized diffusion coefficient D. If the associated transport process proceeds on a time scale comparable to or slower than the period of the temporal oscillation, macroscopic wave propagation phenomena are to be expected, as, for example, realized with the Belousov-Zhabotinsky... 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

In the fabrication of practical E-O devices, all of the three critical materials issues (large E-O coefficients, high stability, and low optical loss) need to be simultaneously optimized. One of the major problems encountered in optimizing polymeric E-O materials is to efficiently translate the large P values of organic chromophores into large macroscopic electro-optic activity (r33). According to an ideal-gas model, macroscopic optical nonlinearity should scale as (M is the chromo-... 

The macroscopic optical responses of a medium are given by its linear and nonlinear susceptibilities, which are the expansion coefficients of the material polarization, P, in terms of the Maxwell fields, 1 3]. For a dielectric or ferroelectric medium under the influence of an applied electric field, the defining equation reads... 

In the past decade, much effort has been made to develop polymeric materials possessing simultaneously large EO coefficients, high thermal and photostabilities, and low optical losses, which could be suitable for incorporation into practical EO devices. One major obstacle that limits the development in this area is to efficiently translate high nonlinearities into large macroscopic EO activities. In fact, although the jijj values of chromophores have been improved more than 250-fold, only several... 

In conclusion from the above formalism we can deduce that the Davydov soliton is the macroscopic envelope of the localized boson condensation of the excitation quanta /) induced by the j8 transformation (3.19). The vanishing of the coefficient G of the nonlinear term of Eq. (3.29), via Eq. (3.13), implies the disappearance of the soliton, which then looks like a necessary consequence of the nonlinearities of the dynamics. 

In the following, macroscopic quantities such as the absorption coefficient, refractive index, and coefficient will be related to molecular concepts such as linear polarizability and nonlinear hyperpolarizability. In the derivation, rather simple models will be employed to demonstrate the basic properties of these quantities. It will become apparent why poling of polymers is essential for second-order nonlinear processes such as second-harmonic generation. The spectral dispersion of the coefficients will also be discussed and examples will be given. 

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