Nonlocal nonlinear response

Several other studies have appeared that are worthy of note. In a series of works by Keller [89-94] and Apell [95], the nonlocal nonlinear response for free-elec-tron-like metals have been examined using various theoretical approaches which are basically extensions of linear theories on the optical response of metals. The results [92] reduce to those obtained by Rudnick and Stern [26] using a similar approach when the free-electron gas is considered to be homogeneous. 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

The generalization of the Fukui functions to nonlinear and nonlocal chemical responses is done in Refs. [26,32] by using N derivatives and the KS perturbation equations. In this section, we propose a brief survey of a complementary derivation based on the concept of the internal charge transfer A introduced above. A more detailed discussion, including computational schemes, will be presented elsewhere. 

In much of the above analysis, the relative magnitude of the surface and bulk contribution to the nonlinear response has not been addressed in any detail. As noted in Section 3.1, in addition to the surface dipole terms of Eq. (3.9), there are also nonlocal electric-quadrupole-type nonlinearities arising from the bulk medium. The effective polarization is made of a combination of surface nonlinear polarization, PNS (2co) (Eq. (3.9)), and bulk nonlinear polarization (Eq. (3.8)) which contains bulk terms y and . The bulk term y is isotropic with respect to crystal rotation. Since it appears in linear combination with surface terms (e.g. Eq. (3.5)), its separate determination is not possible under most circumstances [83, 129, 130, 131]. It mimics a surface contribution but its magnitude depends only upon the dielectric properties of the bulk phases. For a nonlinear medium with a high index of refraction, this contribution is expected to be small since the ratio of the surface contribution to that from y is always larger than se2(2co)/y. The magnitude of the contribution from depends upon the orientation of the crystal and can be measured separately under conditions where the anisotropic contribution of vanishes. 

The nonlinear response at the space-time point (r,t) is due not only to the action of the applied fields at the same space-time point, but also of the fields at a different space-time point (r, t ). In other words, the response is a nonlocal function of time... 

Temporally nonlocal nonlinearities naturally arise if tire response of the medium is not instantaneous and generally will be manifested if short laser pulses are used. Becairse of the delayed reaction of the indirced refractive index changes, a time-dependent phase shift between the intensity and the index grating function will occiu... 

Apart from its presumed nonlinear capabilities, (258) is also more general than the linear response work in preceding sections because it invokes the spatial nonlocality of the uniform gas xc kernel making it more comparable to the work of Dabrowski [102]. To compare it with the discussions above, we now make the local approximation... 

Equation (524) is an exact result for the displacement SA(r, t)) fromequilibriumarising from the external force F (f, t). As evident from Eq. (524), the response of the system to the external force F (r, t) is nonlocal in space and time. Moreover, the response is nonlinear with the nonlinear dependence parried by the nonequilibrimn time correlation function r t ) through pf). 

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