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An analytic dual-mode primary feed is first modeled to illuminate offset reflector antennas to reduce their cross polarization. The feed is linearly polarized and includes the dominant TE11 and higher order TE21 modes. The phase error is then applied to the primary feed to investigate its impact on the cross polarization of offset reflector antennas. Two cases are studied in terms of which of the above-mentioned modes will be affected by the phase errors. The phase errors applying to the resultant dual-mode feed are studied as the first case. The second case includes the effect of phase error only on the higher order TE21 mode resulting in different phase center locations for each mode. Asymmetric right-left phase patterns are used to conduct the study. Both linear and quadratic phase variations are studied. A broad range of focal-length-to-diameter (f/D) ratios from 0.5 to 1.1 are considered to investigate the reduced cross polarization properties in the presence of phase errors. It is shown that the reflector cross polarization increases drastically in the presence of the phase errors. In particular, the boresight-null cross polarization will no longer exist when the phase center location of the second mode is displaced from that of the dominant mode.