The radio intensity and polarization footprint of a cosmic-ray induced extensive air shower is determined by the time-dependent structure of the charge and current distribution residing in the plasma cloud at the shower front. For extracting physics, such as cosmic ray mass or atmospheric electric fields, it is important to determine this charge-current distribution in the plasma cloud, the longitudinal shower structure. To determine the longitudinal shower structure 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 and thus can be used in an chi-square optimization procedure to extract the longitudinal shower structure form a radio footprint, where intensity as well as polarization observables are fitted.
On the basis of air-shower universality we propose a simple parametrization of the radial 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.