  We describe a new approach to calculating photon trajectories and
gravitational lensing effects in the strong gravitational field of the Kerr
black hole. These techniques are applied to explore both the imaging and
spectral properties of photons that perform multiple orbits of the central mass
before escaping to infinity. Viewed at large inclinations, these higher order
photons contribute $\sim 20 %$ of the total luminosity of the system for a
Schwarzschild hole, whilst for an extreme Kerr black hole this fraction rises
to $\sim 60 %$. In more realistic models these photons will be re-absorbed by
the disc at large distances from the hole, but this returning radiation could
provide a physical mechanism to resolve the discrepancy between the predicted
and observed optical/UV colours in AGN. Conversely, at low inclinations, higher
order images re-intercept the disc plane close to the black hole, so need not
be absorbed by the disc if this is within the plunging region. These photons
form a bright ring carrying approximately 10% of the total disc luminosity for
a Schwarzchild black hole. The spatial separation between the inner edge of the
disc and the ring is similar to the size of the event horizon. This is
resolvable for supermassive black holes with proposed X-ray interferometery
missions such as MAXIM, so has the potential to provide an observational test
of strong field gravity.
