Tensor reduced susceptibility

The second-order nonlinear susceptibility tensor ( 3> 2, fOj) introduced earlier will, in general, consist of 27 distinct elements, each displaying its own dependence on the frequencies oip cci2 and = oi 012). There are, however, constraints associated with spatial and time-reversal symmetry that may reduce the complexity of for a given material [32, 33 and Ml- Flere we examine the role of spatial synnnetry. 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization Ps 2a)) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor co, m) is large enough to dwarf the contribution from higher... 

In this model, the structural symmetry of the boundary region is reflected in the form and magnitude of the tensor elements of the surface nonlinear susceptibility, xf and the bulk anisotropic susceptibility, . For 1.06/tm excitation, the penetration depth of E(co) is about 100 A. The surface electric dipole contribution is thought to arise from the first 10 A. The electric quadrupole allowed contribution to E(2to) from the decaying incident field is attenuated by e-2 relative to the surface dipole contribution. Consequently, the symmetry of the SH response should reflect the symmetry of at least the first two topmost layers. For a perfectly terminated (111) surface, the observed symmetry should be reduced from the 6 m symmetry of the topmost layer to 3 m symmetry as additional layers are included. This is consistent with the observations for the centrosymmetric Si(lll) surface response shown in Fig. 3.2 [67, 68]. 

The surface susceptibility is a 27-element tensor that can generally be reduced... 

Kleinman symmetry (index permutation symmetry). Far from resonances of the medium where dispersion is negligible, the susceptibilities become to a good approximation invariant with respect to permutation of all Cartesian indices (without simultaneous permutation of the frequency arguments). This property is called Kleinman symmetry (Kleinman, 1962). It is important in the discussion of the exchange of power between electromagnetic waves in an NLO medium. In many cases approximate validity of Kleinman symmetry can be used effectively to reduce the number of independent tensor components of an NLO susceptibility. 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

The surface susceptibility Xsfg is a 27-element tensor that can generally be reduced to a handful of non-vanishing elements after consideration of the symmetry of the system. In particular, the interface between two isotropic bulk media is isotropic in the plane of the interface (it has Coov symmetry), and Xsfg reduces to the following four independent non-zero elements... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

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 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... 

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]. 

The second derivatives form the reduced magnetic susceptibility tensor tc with components... 

The (reduced) magnetic susceptibility tensor appears in the energy term... 

Then the reduced paramagnetic susceptibility tensor becomes expressed as... 

This reduces to the common van Vleck formula when applied to a diagonal element of the susceptibility tensor... 

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