  The non-commutative electrodynamics based on the canonical Poisson gauge
theory is studied in this paper. For a pure spatial non-commutativity, we
investigate the plane wave solutions in the presence of a constant and uniform
magnetic background field for the classical electrodynamics in canonical
Poisson gauge theory. We obtain the properties of the medium ruled by the
permittivity and the permeability tensors in terms of the non-commutative
parameter, with the electrodynamics equations in the momentum space. Using the
plane wave solutions mentioned, the dispersion relations are modified by the
magnetic background, and the correspondent group velocity is affected by the
spatial non-commutative parameter. We construct the energy-momentum tensor and
discuss the conserved components of this tensor in the spatial non-commutative
case. The birefringence phenomenon is showed through the modified dispersion
relations, that depends directly on the non-commutative corrections and also on
the magnetic background field. Using the bound of the polarized vacuum with
laser (PVLAS) experiment for the vacuum magnetic birefringence, we estimate a
theoretical value for the spatial non-commutative parameter.
