The final beam shape has nothing to do with the shape of the pump diode, Mr. Livinloud. There are square pump diodes that emit a square beam, and yet the green beam is still round. The light doesn't just pass through and change color, it is absorbed by the gain medium and the gain medium emits its own light. A guasian dot (or other TEM modes) is simply a more efficient and more stable use of the cavity. One reason would be the intensity at the corners isn't high enough to cause stimulated emission.