  We investigate the possibility of determining the mass distribution of a
gravitational lens via lensing observations. We consider an extended, compact
gravitational lens, representing its static external gravitational potential
via an infinite set of symmetric trace free (STF) multipole moments. Within the
wave-optical treatment, we evaluate the caustics formed in the lens's point
spread function (PSF). We study the only quantity that is available in
astronomical lensing observations: the image of that PSF formed by an imaging
telescope. This observable may be used to recover some physical characteristics
of the lens, including its shape, orientation and composition. Illustrating
this, we study exotic gravitational lenses formed by several well-known solids
with uniform density. We show that when moments beyond the quadrupole are
observed, some of the symmetry properties of the lens can be recovered. The
presence of an octupole moment implies breaking the "north-south" symmetry of
the mass distribution in the lens. The presence of a rotated hexadecapole
moment implies breaking axial symmetry. As such, if observations of lensed
images allow the reconstruction of these moments, important information about
the mass distribution and dynamics of the lens can be obtained. This may help
with choosing the most appropriate mass profile that is used to characterize
the mass distribution of astrophysical lenses, such as the dark matter halos
that are presumed to contain most of the mass of galaxies and clusters of
galaxies. Our results are novel and offer new insight into gravitational
lensing by realistic astrophysical systems.
