  Dichroic tomography is a 3D imaging technique in which the polarization of
the incident beam is used to induce contrast due to the magnetization or
orientation of a sample. The aim is to reconstruct not only the optical density
but the dichroism of the sample. The theory of dichroic tomographic and
laminographic imaging in the parallel-beam case is discussed as well as the
problem of reconstruction of the sample's optical properties. The set of
projections resulting from a tomographic/laminographic measurement is not
sufficient to reconstruct the magnetic moment for magnetic circular dichroism
unless additional constraints are applied or data are taken at two or more tilt
angles. For linear dichroism, three polarizations or three tilt angles are
required to provide enough information to reconstruct without constraints.
Possible means of applying constraints are discussed. Furthermore, it is shown
that for linear dichroism, the basic assumption that the absorption through a
ray path is the integral of the absorption coefficient, defined on the volume
of the sample, along the ray path, is not correct when dichroism or
birefringence is strong. This assumption is fundamental to tomographic methods.
