Calculation of Doubly Curved Reflectors for Shaped Beams

A method based upon conservation of energy and the simple laws of geometrical optics is described for the calculation of double-curvature surfaces to produce from a point source a shaped beam of arbitrary shape in one plane and uniformly narrow in the perpendicular planes. A specific application of the shaped-beam antenna is in connection with radar antennas for airborne navigational systems, for which the optimum elevation pattern is found empirically to be G(0) = Kcsc²0 cos 0. A reflector to produce from a given primary source the required pattern is the envelope of a family of paraboloids determined by a central-section curve which is adjusted to give the necessary distribution of energy for the shaped beam. A test for the single-valuedness of a computed surface is described. Patterns are shown for experimental antennas whose reflectors were computed by this method. It is demonstrated that some control of the antenna pattern can be achieved by proper motion of the antenna feed. A discussion of errors is included in an Appendix.