  We develop a formalism for the calculation of the macroscopic dielectric
response of composite systems made of particles of one material embedded
periodically within a matrix of another material, each of which is
characterized by a well defined dielectric function. The nature of these
dielectric functions is arbitrary, and could correspond to dielectric or
conducting, transparent or opaque, absorptive and dispersive materials. The
geometry of the particles and the Bravais lattice of the composite are also
arbitrary. Our formalism goes beyond the longwavelenght approximation as it
fully incorporates retardation effects. We test our formalism through the study
the propagation of electromagnetic waves in 2D photonic crystals made of
periodic arrays of cylindrical holes in a dispersionless dielectric host. Our
macroscopic theory yields a spatially dispersive macroscopic response which
allows the calculation of the full photonic band structure of the system, as
well as the characterization of its normal modes, upon substitution into the
macroscopic field equations. We account approximately for the spatial
dispersion through a local magnetic permeability and we analyze the resulting
dispersion relation, obtaining a region of left-handedness.

Optics