Hyperpolarizability tensor media

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 largest component of the Y-tensor is in the conjugation direction. Therefore,.even though no particular bulk symmetry is required for nonzero x > a medium in which all conjugated polymeric chains align in the same direction should have a larger x value along the chain direction relative to that in an amorphous or disordered form of the same polymer. Studies of x in ordered or stretch-oriented polymers as discussed below confirm this prediction. Finally, the polymeric chains should pack as closely as possible in order to maximize the hyperpolarizability density and hence x ... 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

For molecules containing several conjugated bonds yn becomes much larger than y°. Of course, y itself is a fourth rank tensor property (analogous to x(3)) and can be specified in the molecular or laboratory reference frames. For an isotropic medium one measures an orientational average of the hyperpolarizability... 

In this equation, po is the permanent dipole moment of the molecule, a is the linear polarizability, 3 is the first hyperpolarizability, and 7 is the second hyperpolarizability. a, and 7 are tensors of rank 2, 3, and 4 respectively. Symmetry requires that all terms of even order in the electric field of the Equation 10.1 vanish when the molecule possesses an inversion center. This means that only noncentrosymmetric molecules will have second-order NLO properties. In a dielectric medium consisting of polarizable molecules, the local electric field at a given molecule differs from the externally applied field due to the sum of the dipole fields of the other molecules. Different models have been developed to express the local field as a function of the externally applied field but they will not be presented here. In disordered media,... 

This description of quantum mechanical methods for computing (hyper)polarizabilities demonstrates why, nowada, the determination of hyperpolarizabilities of systems containing hundreds of atoms can, at best, be achieved by adopting, for obvious computational reasons, semi-empirical schemes. In this study, the evaluation of the static and dynamic polarizabilities and first hyperpolaiizabilities was carried out at die Time-Dependent Hartree-Fock (TDOT) [39] level with the AMI [50] Hamiltonian. The dipole moments were also evaluated using the AMI scheme. The reliability of the semi-empirical AMI calculations was addressed in two ways. For small and medium-size push-pull polyenes, the TDHF/AMl approach was compared to Hartree-Fock and post Hartree-Fock [51] calculations of die static and dynamic longitudinal first hyperpolarizability. Except near resonance, the TDHF/AMl scheme was shown to perform appreciably better than the ab initio TDHF scheme. Then, the static electronic first hyperpolaiizabilities of the MNA molecule and dimer have been calculated [15] with various ab initio schemes and compared to the AMI results. In particular, the inclusion of electron correlation at the MP2 level leads to an increase of Paaa by about 50% with respect to the CPHF approach, similar to the effect calculated by Sim et al. [52] for the longitudinal p tensor component of p-nitroaniline. The use of AMI Hamiltonian predicts a p aa value that is smaller than the correlated MP2/6-31G result but larger than any of the CPHF ones, which results fi-om the implicit treatment of correlation effects, characteristic of die semi-empirical methods. This comparison confirms that a part of die electron... 

The classic absorption, scattering, reflection, or refraction, the intensity of the light reaching the detector is proportional to the intensity of the incident radiation. When one or more laser beams propagating in materials are large enough, the induced polarization fields are proportional to the product of the incident fields. The polarization p. induced in an atom or a molecule by an external field E can be written as Eq. (1). Where the vectors of p- and E are related by the tensors a, 3, and 7, which are often referred to as the polarizability, hyperpolarizability, and second hyperpolarizability, respectively. Similarly, the polarization induced in a medium by an optical field, can be represented by a power series in the optical fields [Eiq. (2)] where X " is the nth-order susceptibility. 

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