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The Five-hundred-meter Aperture Spherical radio Telescope (FAST) has been fully operational since 11 January 2020. We present a comprehensive analysis of the beam structure for each of the 19 feed horns on FAST's L-band receiver across the Stokes I, Q, U, and V parameters. Using an on-the-fly mapping pattern, we conducted simultaneous sky mapping using all 19 beams directed towards polarization calibrators J1407+2827 and J0854+2006 from 2020 to 2022. Electromagnetic simulations were also performed to model the telescope's beam patterns in all Stokes parameters. Our findings reveal a symmetrical Gaussian pattern in the Stokes I parameter of the central beam without strong sidelobes, while the off-center beams exhibit significant asymmetrical shapes that can be fitted using a combination of log-normal and Gaussian distributions. The inner beams have higher relative beam efficiencies and smaller beam sizes compared to those of the outer beams. The sidelobes of the inner beams contribute approximately 2% of the total flux in the main lobe, increasing to 5% for outer beams, with a peak at 6.8%. In Stokes U, a distinct four-lobed cloverleaf beam squash structure is observed, with similar intensity levels in both inner and outer beams. In Stokes V, a two-lobed beam squint structure is observed in the central beam, along with a secondary eight-lobed structure. The highest squint peak in Stokes V is about 0.3% of the Stokes I in the outer beams. These results align closely with the simulations, providing valuable insights for the design of radio multi-beam observations., Comment: 26 pages, 20 figures. Submitted to AJ

We present the polarization calibration of the 19-beam receiver at 1420 MHz within the full illumination of the Five-hundred-meter Aperture Spherical Telescope from October 2018 to March 2023. We perform spider observations to characterize the on-axis Mueller matrix of the central beam. The calibrated polarization percentage and polarization angle of a source with strong linear polarization emission are about 0.2\% and 0.5$^{\circ}$. Several parameters of the central-beam Mueller matrix show time variability from months to years, suggesting relatively frequent polarization calibrations are needed. We obtain the Mueller matrix parameters of the 18 off-center beams with the combination of on-the-fly observations and spider observations. The polarization calibration provides consistent fractional Stokes parameters of the 19 beams, although the Mueller matrix parameters of the off-center beams are not as accurate as those of the central beam. The Mueller matrix parameters of the central beam do not show a strong dependence on the reflector surface. However, we notice different off-center Mueller matrix parameters between the eastern and western sides of the reflector surface. We provide average parameters of the 19-beam Mueller matrices which should be applicable to observations from 2020 to 2022 with several caveats. After applying the average parameters, on-axis fractional linear polarization measurements $\gtrsim$ 10\% and on-axis fractional circular polarization measurements $\gtrsim$ 1.5\% can be considered high-confidence detections. For sources with weak polarization, timely polarization calibrations using spider observations are required., Comment: 16 pages, 13 figures, , Submitted to AJ