2. Maxwell's Equations

In this chapter we start our investigation of the inverse scattering problem for Maxwell's equations by presenting some basic results for solutions of the time-harmonic Maxwell's equations that will be needed in subsequent chapters. Since the focus of this book is the inverse scattering problem rather than Maxwell's equations in general, we shall make no effort at completeness. In particular, we will provide little or no proof of the results given but instead will refer the reader to either [50] or [93]. For more information on modeling of electromagnetic phenomena see [106]. 2.1 The Scattering of Electromagnetic Waves We begin by considering electromagnetic wave propagation in a source free isotropic medium in with constant electric permittivity , magnetic permeability , and electric conductivity . The electromagnetic wave is then described by the electric field ℰ and the magnetic field ℋ satisfying the time-domain Maxwell's equations curl ℰ+ μ0 ∂ℋ ∂t =0,curl ℋ− ε0 ∂ℰ ∂t = σ0 ℰ.