Reflection and transmission operators for strips or disks embedded in homogenous and layered media

A notation is proposed to simplify the solution of scattering by strips and disks. Vector Fourier transforms are used and a double dot product for inner products in an uncountably infinite dimensional linear vector space is introduced. Scattering by a strip or a disk is characterized using a reflection operator and a transmission operator that relate the continuum of scattered waves to a continuum of incident waves. After the reflection operator for a single strip or disk is derived, it is shown how the reflection operator for a strip or disk in the presence of another reflecting medium, e.g. a layered medium, can be derived. The scattering by N strips or disks in a homogeneous medium is also discussed. The reflection operator for an embedded strip or disk in a layered medium is then derived. The method can be generalized to N strips or disks embedded in a layered medium and to a slot or aperture