Electric field reconstruction with three polarizations for the radio detection of ultra-high energy particles

Accurate reconstruction of the electric field produced by Extensive Air Showers from the signals recorded by the antennas is essential for the radio detection technique, as the key parameters needed to retrieve information about the primary particle that generated the shower are the amplitude, polarization, frequency spectrum and energy fluence carried by the electric field at each measurement position. Conventional electric field reconstruction methods primarily focus on antennas with two horizontal polarizations. In this paper, we introduce an analytical $\chi^2$ minimization method that operates with both two and three polarizations, providing the reconstructed electric field at each antenna. This solution has been verified for simple and realistic antenna responses, with a particular focus on inclined air showers. Our method achieves a standard deviation better than 4\% in determining the peak envelope amplitude of the electric field and better than 6\% in the estimation of the energy fluence, with an antenna response dependent bias. Additionally, we have studied the dependence of the method with arrival direction showing that it has a good performance in almost all of them. This work also demonstrates that incorporating vertically polarized antennas enhances the precision of reconstruction, leading to a more accurate and reliable electric field estimation for inclined air showers. Consequently, the method improves our ability to extract information about cosmic rays from the detected signals in current and future experiments.