Electric Field for Radio Detection of Inclined Air Showers in Three Polarizations

Accurate reconstruction of the electric field produced by extensive air showers is essential for the radio-detection technique, as the key parameters of interest of the primary particles that generated the showers are the amplitude, polarization, frequency spectrum, and energy fluence carried by the electric field at each receiving radio antenna. Conventional electric-field reconstruction methods primarily focus on antennas with two horizontal polarizations. In this work, we introduce an analytic $\chi^2$ minimization method that is applicable to both two and three polarizations. This solution has been verified for simple and realistic antenna responses, with a particular focus on inclined air showers. Our method achieves standard deviations better than 4% and 6% for the estimation of the Hilbert peak envelope amplitude of the electric field and the energy fluence, respectively, 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 the zenith range from 63$^\circ$ up to 80$^\circ$. This work also demonstrates that incorporating vertically polarized antennas enhances the precision of the reconstruction, leading to a more accurate and reliable electric-field estimation for inclined air showers.