Third-order susceptibility tensor

In the case of THG, the third-order susceptibility corresponds to a nonlinear polarization component, which oscillates at the third harmonic frequency of the incident laser beam. Regarding the simphfied case of an isotropic solution, only the element ) third-order susceptibility tensor creates... 

For noncentrosymmetric crystals with a 6 mm point group symmetry, such as wurtzite ZnO, the third-order susceptibility tensor (JCsu) has 21 nonzero... 

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. 

If monochromatic and linearly polarized input fields are considered, the third-order susceptibility can be expressed by its tensor components, Xyjj, =... 

The in (18)-(21) are the nth-order susceptibilities. They are tensors of rank n + with 3 " components. Thus, a second-order susceptibility, is a third-rank tensor with 27 components and the third-order susceptibility, a fourth-rank tensor with 81 components. The number of independent and significant elements is (fortunately) much lower (see p. 131). 

In a subsequent paper, Munn [98] showed that the frequency-dependent local-field tensors accounted for the shift of the poles of the linear and nonlinear susceptibilities from the isolated molecular excitation frequencies to the exciton frequencies. The treatment also described the Davydov splitting of the exciton frequencies for situations where there is more than one molecule per unit cell as weU as the band character or wave-vector dependence of these collective excitations. In particular, the direct and cascading contributions to x contained terms with poles at the molecular excitation energies, but they canceled exactly. Combining both terms is therefore a prerequisite to obtaining the correct pole structure of the macroscopic third-order susceptibility. Munn also demonstrated that this local field approach can be combined with the properties of the effective or dressed molecule and can be extended to electric quadrupole and magnetic dipole nonlinear responses [96]. 

The conceptually simplest NLO property is the electric first dipole hyperpolarizability 13. Nevertheless, it is a challenging property from both the theoretical and experimental side, which is related to the fact that, as third-rank tensor, it is a purely anisotropic property. Experimentally this means that (3 in isotropic media (gas or liquid phase) cannot be measured directly as such, but only extracted from the temperature dependence of the third-order susceptibilities In calculations anisotropic properties are often subject to subtle cancellations between different contributions and accurate final results are only obtained with a carefully balanced treatment of all important contributions. 

Of course, the frequencies and wave vectors fulfil the phase-matching conditions. The third-order susceptibility Xijw is a fourth-rank tensor having a priori 81 elements. In an isotropic material, there remain 21 non-vanishing elements, among which only three are independent [69]. The simplest case consists in a unique incident plane wave, linearly polarized. Indeed, the third-order polarization vector is then parallel to the electric field and reduces to the sum of two propagating terms, one oscillating at the wave circular frequency co, and another at the circular frequency 3(o. The amplitudes of these two contributions write, respectively. 

The original Placzek theory of Raman scattering [30] was in terms of the linear, or first order microscopic polarizability, a (a second rank tensor), not the third order h3q)erpolarizability, y (a fourth rank tensor). The Dirac and Kramers-Heisenberg quantum theory for linear dispersion did account for Raman scattering. It turns out that this link of properties at third order to those at first order works well for the electronically nonresonant Raman processes, but it cannot hold rigorously for the fully (triply) resonant Raman spectroscopies. However, provided one discards the important line shaping phenomenon called pure dephasing , one can show how the third order susceptibility does reduce to the treatment based on the (linear) polarizability tensor [6, 27]. 

Coherent Raman effects originate in the third-order susceptibility fourth-rank tensor) which is the bulk version of the second molecular hyperpolarizability gufifi in (4.1). For the general non-... 

The most widely exploited effect in display applications is the influence of low frequency electric fields on the birefringence. Here, due to a distortion of the director, elements of the dielectric tensor change as a function of the applied field. From Eq. (6), the contribution of the third-order susceptibility -or,0,0,co) to the dielectric tensor is given by e=eo I+) U) Ea Eac]. Although the response is non-local, it is possible to obtain a crude estimate of the average susceptibility E. ), where... 

The situation is simpler for random eollections of moleeules as in, for example, liquids or glasses. As mentioned above, because isotropic media possess a statistical center of symmetry, the second-order susceptibility vanishes. For the third-order susceptibility, only two tensor components, xfni and xfui obtained as a result of orientational averaging. Xnn nan be related to the molecular hyperpolarizability as follows ... 

In Cartesian coordinates obviously there are altogether 3" elements in the third-order susceptibility X > ], a fourth-rank tensor, since i, j, k, /) each has three components 1, 2, 3. In an isotropic medium with inversion symmetry, however, it can be shown that there are only four different components, three of which are independent ... 

The odd order susceptibilities are nonzero in all materials. However, owing to the fact that x is a third rank tensor, the second order susceptibility is nonzero only in noncentrosym-metric materials, that is, materials possessing no center of symmetry. The focus of this paper is on second order processes, and the relationships between the bulk susceptibility, second harmonic generation, and the linear electro-optic effect. For second harmonic generation, Xijl is symmetric in ij, leading to the relationship between the second harmonic coefficient dijk and the bulk second order susceptibility x 2)[i2l... 

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