A Formalism for Calibrating the Instrumental Polarization of Radio Interferometric Arrays at Meter Wavelengths using Unpolarized Sky: A Demonstration using the MWA Observations

Calibration of instrumental polarization is critical for measuring polarized radio emissions from astrophysical sources to extract the magnetic field information in astrophysical, heliospheric, and terrestrial plasmas. At meter wavelengths, calibration of radio polarimetric observations is particularly challenging because of the scarcity of bright polarized sources due to significant Faraday depolarization. Here, we present a novel formalism for polarization calibration using an unpolarized sky model. The formalism is specifically designed for wide-field, low-frequency instruments like the Murchison Widefield Array (MWA), the LOw Frequency ARray (LOFAR), New Extension in Nan\c{c}ay Upgrading LoFAR (NenuFAR), Owens Valley Radio Observatory - Long Wavelength Array (OVRO-LWA), low-frequency telescope of the Square Kilometre Array Observatory (SKAO-low), etc. By leveraging the apparent polarization of the unpolarized sky induced by the polarized primary beam of the radio telescope, this method avoids dependence on bright polarized calibrators. It is also immune to ionospheric Faraday rotation. The validation of the approach via MWA observations confirms the accuracy of the method. This formalism provides a robust framework for low-frequency polarization calibration. It addresses the longstanding calibration challenges and advances the field of low-frequency polarimetry by enabling polarization studies of astrophysical radio sources.