The Ewald–Oseen Extinction Theorem and the Extended Boundary Condition Method

Abstract

Mathematical statements of the Ewald–Oseen extinction theorem have been formulated for a linear homogeneous medium with frequency-domain constitutive relations \(\underline{D} =\underline{\underline{\varepsilon }} \mbox{ $^{\bullet }$ }\underline{E} + [\underline{\underline{\xi }} + \left (\underline{K}-\underline{\varGamma }\right ) \times \underline{\underline{ I}}]\mbox{ $^{\bullet }$ }\underline{H}\) and \(\underline{B} =\underline{\underline{\mu }} \mbox{ $^{\bullet }$ }\underline{H} - [\underline{\underline{\xi }} -\left (\underline{K}+\underline{\varGamma }\right ) \times \underline{\underline{ I}}]\mbox{ $^{\bullet }$ }\underline{E}\). During the formulation, four dyadic Green functions were set up and their characteristics elucidated. The Ewald–Oseen extinction theorem has been exploited to set up the conceptual framework of the extended boundary condition method (EBCM) for scattering by a three-dimensional object made of a homogeneous medium. Implementation of the EBCM is also premised on the availability of bilinear expansions of the dyadic Green functions for both the scattering medium and the medium surrounding the scatterer.