Recently a theory for artificial magnetism has been proposed for photonic crystals in specific geometries allowing a two-dimensional reduction
(see [1-3, 5]).
Here we propose a full 3D generalization: the diffraction of a finite 3D- dielectric crystal is considered at a fixed wavelength
and a limit analysis as the period tends to zero is performed. We evidence a new
microscopic vectorial spectral problem which accounts the resonances of the crystal. The artificial magnetism is then described by a frequency dependent effective permeability tensor with possibly negative eigenvalues.
This suggests that photonic crystals made of dielectric inclusions could be an interesting alternative to the very popular metallic split-ring structure proposed by Pendry  and studied mathematically in [5,6]. Numerical simulations will be presented.
Joined work with D. Felbacq (GES, Montpellier) and
C.Bourel (Imath, Toulon).