  In this paper, we introduce an alternative representation of the
electromagnetic field scattered from a homogeneous sphere coated with a
homogeneous layer of uniform thickness. Specifically, we expand the scattered
field using a set of modes that are independent of the permittivity of the
coating, while the expansion coefficients are simple rational functions of the
permittivity. The theory we develop represents both a framework for the
analysis of plasmonic and photonic modes and a straightforward methodology to
design the permittivity of the coating to pursue a prescribed tailoring of the
scattered field. To illustrate the practical implications of this method, we
design the permittivity of the coating to zero either the backscattering or a
prescribed multipolar order of the scattered field, and to maximize an electric
field component in a given point of space.
