Hyperpolarizability tensors, nonlinear optical

For dipolar chromophores that are the subject of this chapter, only one component of the molecular hyperpolarizability tensor, Pzzz, is important. Thus, the summation in Eq. (8) disappears. Electric field poling induces Cv cylindrical polar symmetry. Assuming Kleinman [12] symmetry, only two independent components of the macroscopic second-order nonlinear optical susceptibility tensor... 

The coeflScients a, P, and y are the second, third, and fourth rank tensors and are referred to as the polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. The hyperpolarizability terms are responsible for the nonlinear response of the molecule to impinging radiation. These coefiBcients are not very large, and the associated nonlinear optical effects are usually studied by taking advantage of the high optical field obtainable with laser beams. 

Nonlinear optical techniques (SHG, SFG) Adsorption kinetics, interfacial coverage, reactioii kinetics, phase transitions, orientational order (average tilt angle), surface chirality. Intensity of the signal reflects the combined effect of interfacial coverage and orientational order. Tilt angles only obtainable if all non-zero elements of the hyperpolarizability tensor can be determined. 

To understand and optimize the electro-optic properties of polymers by the use of molecular engineering, it is of primary importance to be able to relate their macroscopic properties to the individual molecular properties. Such a task is the subject of intensive research. However, simple descriptions based on the oriented gas model exist [ 20,21 ] and have proven to be in many cases a good approximation for the description of poled electro-optic polymers [22]. The oriented gas model provides a simple way to relate the macroscopic nonlinear optical properties such as the second-order susceptibility tensor elements expressed in the orthogonal laboratory frame X,Y,Z, and the microscopic hyperpolarizability tensor elements that are given in the orthogonal molecular frame x,y,z (see Fig. 9). 

The oriented gas model was first employed by Chemla et al. [4] to extract molecular second-order nonlinear optical (NLO) properties from crystal data and was based on earlier work by Bloembergen [5]. In this model, molecular hyperpolarizabilities are assumed to be additive and the macroscopic crystal susceptibilities are obtained by performing a tensor sum of the microscopic hyperpolarizabilities of the molecules that constitute the unit cell. The effects of the surroundings are approximated by using simple local field factors. The second-order nonlinear response, for example, is given by... 

TDHF [52, 53] is one of the most widely-employed ab initio techniques to evaluate nonlinear-optical response tensors. The fDHF approach is size consistent but cannot account for the finite lifetime of the excited states. The matrices of the TDHF equation are expanded in a Taylor series of the perturbation due to the static and/or dynamic electric fields and are solved for each order [52, 53], The so-obtained successive field-derivatives of the density matrix are then inserted into the expressions for the hyperpolarizability,... 

It is most important to note that in many cases of harmonic emission, a more completely index-symmetric form of the polarizability tensor is implicated. Consider once again the prototypical example of optical nonlinearity afforded by harmonic generation. When any harmonic is generated from a plane-polarized beam, in an isotropic medium, it produces photons with the same polarization vector as the incident light. In such a case the radiation tensor pyk becomes fully index-symmetric, and arguments similar to those given above show that only the fully index-symmetric part of the hyperpolarizability tensor, 3p(—2m co, co), can be involved. This does not mean that the tensor itself is inherently fully index-symmetric, but it does mean that experiments of the kind described cannot determine the extent of any index antisymmetry. 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization PfL(2 >) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor x (2 > >, a>) is large enough to dwarf the contribution from higher orders of the multipole expansion like the electric quadrupole contribution and is therefore the simplest approximation for the nonlinear polarization. At pure solvent interfaces, this may not be the case, since the nonlinear optical activity of solvent molecules like water, 1,2-dichloroethane (DCE), alcohols, or alkanes is rather low. The magnitude of the molecular hyperpolarizability of water, measured by DC electric field induced second harmonic... 

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). 

The study of amphiphile ordering at interfaces is necessary to understand many phenomena, like microemulsions, foams or interfacial reactivity. It is expected that the preferential orientation taken by these compounds at interfaces is entirely determined by their interactions with the two solvants forming the interface and the intermolecular repulsion or attraction within the monolayer. As mentioned above, the SH response at liquid/liquid interfaces is dominated by electric dipole contributions and is therefore surface specific. Neglecting the contribution from the sol-vant molecules, which usually only have a weak nonlinear optical activity, the passage from the macroscopic susceptibility tensor xP to the microscopic molecular hyperpolarizability p of the adsorbate is obtained by merely taking the SHG response of the amphiphile monolayer as the superposition of the contribution from each single moiety. Hence, it yields... 

Finally, fhe application of higher order nonlinear optical processes such as third-harmonic generation (Berkovic 1995 Tsang 1995) or fourth-harmonic generation (Lee et al. 1997) could provide even more detailed interface information. For example, in the case of fourth-harmonic generation the induced polarization is given by Pj 4u>) = j k,l,m,nx]tL Ek u>)Ei co)Em u>)E io), i.e., the hyperpolarizability is a tensor of rank 5. Hence if is possible fo resolve up to five-fold surface symmetries. However, the absolute values are... 

The summation runs over repeated indices, /r, is the i-th component of the induced electric dipole moment and , are components of the applied electro-magnetic field. The coefficients aij, Pijic and Yijki are components of the linear polarizability, the first hyperpolarizability, and the second hyperpolarizability tensor, respectively. The first term on the right hand side of eq. (12) describes the linear response of the incident electric field, whereas the other terms describe the nonhnear response. The ft tensor is responsible for second order nonlinear optical effects such as second harmonic generation (SHG, frequency AotAAin, frequency mixing, optical rectification and the electro-optic effect. The ft tensor vanishes in a centrosymmetric envirorunent, so that most second-order nonlinear optical materials that have been studied so far consists of non-centrosyrmnetric, one-dimensional charge-transfer molecules. At the macroscopic level, observation of the nonlinear optical susceptibility requires that the molecular non-symmetry is preserved over the physical dimensions of the bulk stmcture. 

A few further general remarks are in order at this stage. One is to note the fact that the sum over intermediate molecular states, as in Eqs. (76) and (79), in principle applies not only to electronic but also to vibrational levels. Although this issue initially received most attention in connection with molecular hyperpolarizabilities [44], it applies equally to other optical response tensors. The vibrational contributions, which were previously largely overlooked, have now been extensively studied and shown to be important in many applications [45,46]. Second, the polarizabilities associated with nonlinear parametric processes may in most circumstances be regarded as properties of the ground-state molecule, since it is the molecular ground state that usually constitutes the... 

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