The scattering of electromagnetic waves by a perfectly conducting infinite cylinder

Abstract

We consider the scattering of a plane time-harmonic electromagnetic wave by a perfectly conducting infinite cylinder with axis in the direction k, where k is the unit vector along the z axis. Suppose the incident wave propagates in a direction perpendicular to the cylinder. For a given observation angle 0, let F,(O, a)k be the far-field pattern of the electric field corresponding to an incident wave with direction angle a polarized perpendicular to the z axis and let FN(O;a)k be the far-field pattern of the magnetic field corresponding to an incident wave with direction angle a polarized parallel to the z axis. Let {am}:=, be a distinct set of angles in [ -n, 2 3 and p a complex number. Then, necessary and sufficient conditions are given for the set { (1 -p)F& a,) + pFN(& aJ}mm= I to be complete in L z [ -n, n]. Applications, together with numerical examples, are given to the inverse scattering problem of determining the shape of the cylinder from a knowledge of the far-field data.

The use of polarization effects in electromagnetic inverse scattering problems

“…Indeed, it follows from [S] that this new method for solving the inverse scattering problem fails if the wave number is equal to a Maxwell eigenvalue of the scattering obstacle. In an effort to overcome this defect in a special case, Colton and Monk considered the twodimensional scalar case and showed that if the data were chosen to be a convex linear combination of the electric and magnetic far-field patterns then, if this combination was not degenerate, there was no longer any problem with interior eigenvalues [6]. The purpose of this paper is to extend this result to the full three-dimensional vectorvalued problem.…”

“…These two problems are to be understood as problems in constrained optimization where h and r are subject to a priori constraints. For details of this procedure for solving the inverse scattering problem in the (scalar) two-dimensional case, including numerical examples, we refer the reader to [6]. It follows from the proof of Theorem 2.3 that the (unconstrained) cost functional of the sum of the above optimization schemes will have zero as its greatest lower bound provided there exists an electric Herglotz field Eh such that (2.22) is approximately satisfied in the space of square-integrable tangential vector fields defined on aD where E, H satisfy (2.21).…”