The factorization method for Maxwell's equations

Abstract

This chapter examines the time-harmonic Maxwell's equations for isotropic and non-magnetic but possibly inhomogeneous media. It begins by formulating the direct scattering problem for arbitrary measurable and essentially bounded permittivities. It derives an integro-differential equation of Lippmann-Schwinger type, proving equivalence with the weak formulation and the Fredholm property. As in the previous chapters, the chapter introduces the far field patterns and the far field operator, derives some of the most important properties, and formulates the inverse scattering problem. The general results of Chapters 1 and 2 are then applied, and a characterization of the contrast in terms of a Picard series is derived involving only known quantities. For the scalar case of Chapter 4, certain critical values of the frequency have to be excluded, which are eigenvalues of an interior transmission eigenvalue problem for Maxwell's equations. It is shown that there exists a quantifiable number of these values.