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A novel hybrid combination of an analytical asymptotic high-frequency method with a numerical physical optics (PO) procedure is developed to efficiently and accurately predict the far zone fields of extremely long, scanning, very high gain, offset cylindrical reflectors of arbitrary cross-section, with large stacked finite periodic linear phased array feeds, for spaceborne applications. In this method, the field generated by each finite length linear feed array is represented as a spectral integral and the induced current on the cylindrical reflector surface due to this illumination is obtained via the PO approximation. The reflector surface is divided into thin, long, piecewise planar strips along the generator of the cylindrical reflector, and the radiation integral over each strip is evaluated asymptotically in closed form, yielding an eight-term ray solution for the radiated fields. What remains is simply the superposition of the contributions of each strip and linear array in the feed stack. The proposed approach is shown to be extremely efficient and accurate as compared to the conventional PO integration technique. In addition, the method is sufficiently versatile to account for the reflector edge treatments (e.g., using resistive cards), as well as to account for a twist in the reflector surface due to thermal distortions in space.