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An analytic model for the primary feed of offset reflector antennas is presented that facilitates a more general investigation of the cross polarization reduction in both asymmetric and inter-cardinal planes. It includes both first and second order azimuthal modes, primarily the TE11 and TE21 type modes, and allows selection of symmetric and asymmetrical patterns, with different field tapers in the principal planes. Using this model first the influence of the second order TE21 mode on the cross polarization of an offset reflector with a circular rim is investigated. Then, the problem for an offset reflector with an elliptic rim is also studied. It is shown that minimizing the cross polarization in the asymmetric plane does not necessarily reduce it globally, as in the inter-cardinal plane it remains high. The procedure for reduction of the cross polarization globally and its effects on the sidelobe levels of the reflector antenna are also investigated. With the proposed feed model, the aperture efficiency of the reflector antenna is improved compared with a standard Gaussian feed with the same edge taper. Based on this study, a dual-mode circular waveguide horn is also proposed and designed. Its performance on an offset reflector antenna is studied and shown that the cross polarization levels are reduced quite satisfactorily within a respectable frequency band.