  The radio intensity and polarization footprint of a cosmic-ray induced
extensive air shower is determined by the time-dependent structure of the
current distribution residing in the plasma cloud at the shower front. In turn,
the time dependence of the integrated charge-current distribution in the plasma
cloud, the longitudinal shower structure, is determined by interesting physics
which one would like to extract such as the location and multiplicity of the
primary cosmic-ray collision or the values of electric fields in the atmosphere
during thunderstorms. To extract the structure of a shower from its footprint
requires solving a complicated inverse problem. For this purpose we have
developed a code that semi-analytically calculates the radio footprint of an
extensive air shower given an arbitrary longitudinal structure. This code can
be used in a optimization procedure to extract the optimal longitudinal shower
structure given a radio footprint. On the basis of air-shower universality we
propose a simple parametrization of the structure of the plasma cloud. This
parametrization is based on the results of Monte-Carlo shower simulations.
Deriving the parametrization also teaches which aspects of the plasma cloud are
important for understanding the features seen in the radio-emission footprint.
The calculated radio footprints are compared with microscopic CoREAS
simulations.
