  High resolution single-shot non-relativistic ultrafast electron
microscopy(UEM) relies on adaptive optics to compress high intensity bunches
using Radio Frequency (RF) cavities. We present a comprehensive discussion of
the analytic approaches available to characterize bunch dynamics as an electron
bunch goes through a longitudinal focal point after an RF cavity where space
charge effects can be large. Methods drawn from the Coulomb explosion
literature, the accelerator physics literature and the analytic Gaussian model
developed for UEM are compared, utilized and extended in some cases. In
particular the longitudinal focus may occur in two different regimes, a
bounce-back regime and a crossover regime; and we characterize the critical
point separating these regimes in the zero-emittance model. Results from
N-particle simulations using efficient multipole methods are compared to the
theoretical models revealing features requiring extensions of the analytic
approaches; and in particular mechanisms for emittance growth and transfer are
discussed.
