  We propose a theory for a type of apertureless scanning near field microscopy
that is intended to allow the measurement of magnetism on a nanometer length
scale. A scanning probe, for example a scanning tunneling microscope (STM) tip,
is used to scan a magnetic substrate while a laser is focused on it. The
electric field between the tip and substrate is enhanced in such a way that the
circular polarization due to the Kerr effect, which is normally of order 0.1%
is increased by up to two orders of magnitude for the case of a Ag or W tip and
an Fe sample. Apart from this there is a large background of circular
polarization which is non-magnetic in origin. This circular polarization is
produced by light scattered from the STM tip and substrate. A detailed retarded
calculation for this light-in-light-out experiment is presented.
