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A rigorous TE solution to the dielectric wedge antenna fed by a slab waveguide of the same material is presented. The method of solution involves modeling the wedge as a sequence of step discontinuities and uses an iterative procedure to track forward and backward partial wave fields, expressed as modal expansions, to obtain the rigorous field solution. Radiation patterns of directive gain are presented. All patterns smoothly decrease from a maximum in the endfire direction and exhibit extremely low side lobe levels. Longer length wedges or smaller dielectric constant materials are shown to produce higher directivity and smaller half-power beamwidths. For slender, gradually tapered wedges, the reflection coefficient of the guided (surface) wave at the input to the wedge is very small indicating a low VSWR for tapered dielectric antennas and there appears to be no gain limitation with antenna length for these antennas.