Structure of electromagnetic fields in spatially dispersive media of arbitrary geometry

The nature of the electromagnetic field in a spatially dispersive medium, occupying an arbitrary domain V is investigated, under conditions when spatial dispersion effects arise from the presence of an isolated exciton transition band. It is shown that the electric field at frequency ω close to the exciton transition frequency may, in general, be expressed in the form E→ (r→,ω)=E→t(1) (r→, ω)+E→t(2)(r→, ω)+E→l(r→, ω), where E→t(j)(r→, ω)(j=1, 2) are transverse fields and E→l(r→, ω) is a longitudinal field; and that each of these three fields satisfies a Helmholtz equation. The wave numbers occurring in the three Helmholtz equations are the roots of the dispersion relations appropriate to the medium. It is further shown that the three fields are coupled by a linear relation, which is shown to imply a recently derived nonlocal boundary condition on the nonlocal polarization, expressed in the form of an extinction theorem. These results are generalizations of certain results obtained not long ago by Sein, Birman and Sein, Agarwal, Pattanayak, and Wolf, and Maradudin and Mills