Monotonicity in inverse scattering for Maxwell's equations

Abstract

<p style='text-indent:20px;'>A recent area of research in inverse scattering theory has been the study of monotonicity relations for the eigenvalues of far field operators and their use in shape reconstruction for inverse scattering problems. We develop such monotonicity relations for an electromagnetic inverse scattering problem governed by Maxwell's equations, and we apply them to establish novel rigorous characterizations of the shape of scattering objects in terms of the corresponding far field operators. Along the way we establish the existence of electromagnetic fields that have arbitrarily large energy in some prescribed region, while at the same time having arbitrarily small energy in some other prescribed region. These localized vector wave functions not only play an important role in the proofs of the novel monotonicity based shape characterizations but they are also of independent interest. We conclude with some simple numerical demonstrations of our theoretical results.</p>