Localized nonlinear approximation

We will demonstrate now that the TQA approximation can be treated as a modification of the extended Born (localized nonlinear - LN) approximation, introduced above. Let us rewrite equation (9.96) in the form 

Taking into account once again the fact that the Green s tensor (r r) exhibits either a singularity or a peak at the point where = r, one can calculate the Born approximation Gg [A5 (r) E (r)] using the formula 

This approximation is particularly appropriate if the background field is a smoothly varying function of the coordinates it forms the basis of localized nonlinear approximations (Habashy et al., 1993). 

Formulae (9.107) and (9.108) give the localized nonlinear (LN) approximation introduced by Habashy et al. (1993), and the tensor 

we can see that the difference between the tensor quasi-analytical approximation and the localized nonlinear approximation is determined by a term  

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This constrained nonlinear optimisation problem can be solved using a Successive Quadratic Programming (SQP) algorithm. In the SQP, at each iteration of optimisation a quadratic program (QP) is formed by using a local quadratic approximation to the objective function and a linear approximation to the nonlinear constraints. The resulting QP problem is solved to determine the search direction and with this direction, the next step length of the decision variable is specified. See Chen (1988) for further details. 

Note that approximation (9.76) is often referred to as a localized nonlinear (LN) approximation. 

We shall analyze in the next sections the different techniques of the TQL equation solution hich result in different analytical expressions for the electrical reflectivity tensor A (r ). In particular, one of these solutions gives rise to the extended Born, or localized nonlinear (LN) approximation, introduced above. Note that these approximations may be less accurate than the original QL approximation with a fine grid for the discretization of A(rj). 

The LQL approximation has a similar background to the localized nonlinear (LN) approximation, but there are some important differences. The LN approximation also replaces the total field inside inhomogeneity with a product of the background field and a scattering tensor E (r),... 

Another scheme to calculate and interpret macroscopic nonlinear optical responses was formulated by Mukamel and co-workers [112 114] and incorporated intermolecular interactions as well as correlation between matter and the radiation field in a consistent way by using a multipolar Hamiltonian. Contrary to the local field approximation, the macroscopic susceptibilities cannot be expressed as simple functionals of the single-molecule polarizabilities, but retarded intermolecular interactions (polariton effects) can be included. 

Density Functional Theory and the Local Density Approximation Even in light of the insights afforded by the Born-Oppenheimer approximation, our problem remains hopelessly complex. The true wave function of the system may be written as i/f(ri, T2, T3,. .., Vf ), where we must bear in mind, N can be a number of Avogadrian proportions. Furthermore, if we attempt the separation of variables ansatz, what is found is that the equation for the i electron depends in a nonlinear way upon the single particle wave functions of all of the other electrons. Though there is a colorful history of attempts to cope with these difficulties, we skip forth to the major conceptual breakthrough that made possible a systematic approach to these problems. 

When Da < 1, all the modes in the sum decay in time towards the C = 0 state, but for Da > 1 at least the mode with n = 0 grows, and (7 = 0 becomes unstable to small localized perturbations. To obtain a nonlinear approximation of the steady filament solution close to the bifurcation point Dac = 1, a perturbation approximation can be used by introducing... 

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

X yxzx — x s xxzx s zxx x s zzz l l e normal to the interface being taken along the Z axis with the X axis in the interface plane. The surface nonlinear polarization Ps (2co) is localized at the interface and is usually described as a nonlinear polarization sheet. This approximation holds because the thickness corresponding to the physical region of the... 

Thus, since intramolecular bonding interactions in the solid are much stronger than relatively weak i n termo1ecu1 ar van der Vaals interactions, each molecular unit is essentially an independent source of nonlinear response, arrayed in an acentric cystal structure, and coupled to its neighbors mainly through weak local fields. In the rigid lattice. gas approximation, the macroscopic susceptibility X is expressed as... 

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