  Converging cylindrical electromagnetic fields in vacuum have been shown (E.
I. Zababakhin, M. N. Nechaev, {\it Soviet Physics JETP}, {\bf 6}, 345 (1958))
to exhibit amplitude "cumulation". It was found that the amplitude of
self-similar waves increases without bounds at finite distances from the axis
on the front of the fields reflected from the cylindrical axis. In the present
paper we propose to exploit this cylindrical cumulation process as a possible
new path towards the generation of ultra-strong electromagnetic fields where
nonlinear quantum electrodynamics (QED) effects come into play. We show that
these effects, as described in the long wave-length limit within the framework
of the Euler Heisenberg Lagrangian, induce a radius-dependent reduction of the
propagation speed of the cumulation front. {Furthermore we compute the
$e^+$-$e^-$ pair production rate at the cumulation front and show that the
total number of pairs that are generated scales as the sixth power of the field
amplitude.
