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A design method is described for the creation of line sources that will yield antenna patterns consisting of a narrow main beam and low sidelobes, with the sidelobe levels different on the two sides of the main beam. The requisite aperture distribution is shown to be symmetrical in amplitude and asymmetrical in phase. Main beam offset and the characteristics of the associated difference pattern are discussed. The boresight error voltage curve is also displayed. Incremental gain resulting from asymmetric sidelobes is analyzed and data is given for typical cases. Experimental confirmation is, presented.