  Electromagnetic properties of periodic two-dimensional sub-wavelength
structures consisting of closely-packed inclusions of materials with negative
dielectric permittivity $\epsilon$ in a dielectric host with positive
$\epsilon_h$ can be engineered using the concept of multiple electrostatic
resonances. Fully electromagnetic solutions of Maxwell's equations reveal
multiple wave propagation bands, with the wavelengths much longer than the
nanostructure period. It is shown that some of these bands are described using
the quasi-static theory of the effective dielectric permittivity
$\epsilon_{qs}$, and are independent of the nanostructure period. Those bands
exhibit multiple cutoffs and resonances which are found to be related to each
other through a duality condition. An additional propagation band characterized
by a negative magnetic permeability develops when a magnetic moment is induced
in a given nano-particle by its neighbors. Imaging with sub-wavelength
resolution in that band is demonstrated.
