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The taper profile of optimized dielectric-rod and horn antennas is synthesized as a series of non-interacting planar radiating apertures. The method is semi-empirical, straightforward to apply, enables the dielectric-rod antenna to be satisfactorily optimized and provides a means of evaluating and optimizing a dielectric-horn antenna with variable wall thickness. The optimum profiles are taken as those which smoothly transform the surface-wave power from the launcher to the radiating aperture. The optimization of the dielectric-rod antenna considerably improves the radiation pattern while computations supported by measurements confirm earlier reports that a dielectric-horn can have a higher gain than a metal horn of similar dimension but side-lobe level is seen to be an important issue. Wide flare-angle horns give ideal E-plane patterns at the expense of a high side-lobe level in the H-plane; for small flare angles the dielectric horn gives similar patterns to the tapered rod antenna and thus preserves rotational symmetry. Calculations throughout are restricted to cylindrical geometry but other geometries and variations on the dielectric-horn principles are described. Useful engineering design data have been compiled for both the dielectric-horn and rod antennas and curves are given which determine near-optimum parameters for gains up to about 20 dB which is seen to be a practical operating limit for these surface wave devices. A unified impression of dielectric antennas emerges with the important conclusion that, when optimized, dielectric-rod and horn antennas are in fact competitive with small metal horns for some applications; furthermore the dielectric-horn antenna, used singly or in arrays, is an ideal device for producing a low side-lobe level in the E-plane.