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Dielectric resonator antennas (DRAs) offer a number of advantages, amongst which is offering a wider bandwidth than the microstrip patch antennas commonly used at the same frequency. The shape of DRAs can take different forms, such as rectangular, hemispherical and cylindrical. We rigorously study the aperture-coupled prolate hemispheroidal DRA excited at the fundamental broadside mode. An analytical solution for the radiation by a prolate hemispheroidal DRA using a dyadic Green's function technique is presented. Compared with hemispherical DRAs, hemispheroidal DRAs add one more freedom of design in terms of different semiaxes dimensions and may lead to a simple feed mechanism for generating circularly polarized waves. Another parameter that may lead to better results is the greater diversity of mode excitation than hemispherical DRAs have. We consider a prolate hemispheroidal DRA excited by a slot aperture. The analytical development of the solution together with various numerical results are presented and discussed.