A rigorous solution for the TB and TM polarization 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 using 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 very low side lobes. Longer length wedges are shown to produce higher directivity and smaller half-power beamwidths (HPBW). For TB polarization, wedges with larger values of dielectric constant yield smaller directivity and broader HPBW. In contrast, for TM polarization, wedges with larger values of dielectric constant have essentially same directivity and HPBW as wedges of smaller dielectric constant material. For TB or TM polarization, slender, gradually tapered wedges, yield a reflection coefficient of the guided surface wave on the feed guide at the base of the antenna that is very small. This indicates that tapered dielectric antennas produce low VSWR values. In addition, there appears to be no gain limitation with antenna length for these antennas. Frequency characteristics are examined to show that the dielectric wedge antenna is a broadband antenna. The method of solution is general and can be applied to a broad class of dielectric antennas having different geometries.