  In a recent paper (Brown & Battye 2011), we proposed the use of integrated
polarization measurements of background galaxies in radio weak gravitational
lensing surveys and investigated the potential impact on the statistical
measurement of cosmic shear. Here we extend this idea to reconstruct maps of
the projected dark matter distribution, or lensing convergence field. The
addition of polarization can, in principle, greatly reduce shape noise due to
the intrinsic dispersion in galaxy ellipticities. We show that maps
reconstructed using this technique in the radio band can be competitive with
those derived using standard lensing techniques which make use of many more
galaxies. In addition, since the reconstruction noise is uncorrelated between
these standard techniques and the polarization technique, their comparison can
serve as a powerful check for systematics and their combination can reduce
noise further. We examine the convergence reconstruction which could be
achieved with two forthcoming facilities: (i) a deep survey, covering 1.75
square degrees using the e-MERLIN instrument currently being commissioned in
the UK and (ii) the high resolution, deep wide field surveys which will
eventually be conducted with the Square Kilometre Array.
