An improved-efficiency quasi-optical antenna for the transformation of the output of a gyrotron operating in a whispering-gallery mode has been reported by M. Iima et al. (1989). The improved antenna incorporates a shaped reflecting surface attached to the helically-cut waveguide launcher of the configuration proposed by S.N. Vlasov et al. (1975). This second reflector allows for the manipulation of the output-beam power distribution parallel to the direction of the polarization of the beam. A measured Gaussian-like near-field profile in which 93% of the beam power is contained in the main lobe has been obtained. The present authors establish, in a rigorous theoretical manner, a dual-reflector synthesis technique based upon geometrical optics principles to obtain the shapes of the reflectors. The design procedure may be extended to allow for manipulation of the power distribution across both dimensions of the output beam.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>