  The electromagnetic modes and the resonances of homogeneous, finite size,
two-dimensional bodies are examined in the frequency domain by a rigorous full
wave approach based on an integro-differential formulation of the
electromagnetic scattering problem. Using a modal expansion for the current
density that disentangles the geometric and material properties of the body the
integro-differential equation for the induced surface (free or polarization)
current density field is solved. The current modes and the corresponding
resonant values of the surface conductivity (eigen-conductivities) are
evaluated by solving a linear eigenvalue problem with a non-Hermitian operator.
They are inherent properties of the body geometry and do not depend on the body
material. The material only determines the coefficients of the modal expansion
and hence the frequencies at which their amplitudes are maximum (resonance
frequencies). The eigen-conductivities and the current modes are studied in
detail as the frequency, the shape and the size of the body vary. Open and
closed surfaces are considered. The presence of vortex current modes, in
addition to the source-sink current modes (no whirling modes), which
characterize plasmonic oscillations, is shown. Important topological features
of the current modes, such as the number of sources and sinks, the number of
vortexes, the direction of the vortexes are preserved as the size of the body
and the frequency vary. Unlike the source-sink current modes, in open surfaces
the vortex current modes can be resonantly excited only in materials with
positive imaginary part of the surface conductivity. Eventually, as examples,
the scattering by two-dimensional bodies with either positive or negative
imaginary part of the surface conductivity is analyzed and the contributions of
the different modes are examined.
