Spatial dispersion in superlattices

Below we assume that superlattice layers 1 or 2 can be of different natures and can be, for example, organic or inorganic. In the vicinity of excitonic resonances even at 12 3 / ( is the Bohr radius of an exciton, of the order of the lattice constant in the case of Frenkel excitons) the nonlocality of the dielectric permeability can be taken into account. Consider, for instance, the frequency 

When nonlocality is not taken into account, the value of the x-component of dielectric displacement vector averaged over the superlattice period is (see also Section 8.1) 

In the nonlocal case, instead of (8.12) and (8.13) one should write, respectively  

To proceed we have to express the x-component of the polarization in layer 1 in terms of Ex and its z-component in terms of Dz. We can obtain these relations if we take into account that the displacement vector by definition is D = e i(w, A )E = eooE + 47rP. For the dielectric tensor in layer 1 we can take the 

Analogously, using E = D/ei(w, K) we can find from the last relation that 

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