  A method is described to solve the Poisson problem for a three dimensional
source distribution that is periodic into one direction. Perpendicular to the
direction of periodicity a free space (or open) boundary is realized. In beam
physics, this approach allows to calculate the space charge field of a
continualized charged particle distribution with periodic pattern.
  The method is based on a particle mesh approach with equidistant grid and
fast convolution with a Green's function. The periodic approach uses only one
period of the source distribution, but a periodic extension of the Green's
function.
  The approach is numerically efficient and allows the investigation of
periodic- and pseudo-periodic structures with period lengths that are small
compared to the source dimensions, for instance of laser modulated beams or of
the evolution of micro bunch structures. Applications for laser modulated beams
are given.
