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Using a simple polarization picture of a medium in which there is spatial dispersion and using the integral equation method of Ewald-Oseen, some rigorous results are obtained for the optics of a bounded spatially dispersive medium. In particular, an extinction theorem for polaritons, and rigorous additional boundary conditions are obtained. The normal-incidence propagation of transverse modes for an isotropic crystal bounded by a plane is analyzed. The rigorous boundary conditions obtained by this method, in general, do not agree with other popular choices, based upon the same constitutive relations.
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Birman et al. (1972) studied this question.
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