Optimization criterion for illuminating circular antenna apertures

For many applications the ideal antenna pattern is one in which radiated power is spread uniformly over the cone
\theta < \theta_{0}
, with no power going outside that cone. For example, either the target orientation or the antenna axis may only be known with tolerance
\theta_{0}
. While this ideal pattern is unachievable by a finite antenna aperture, it may be asymptotically approached as aperture size goes to infinity. We here formulate a new worst-case criterion for comparing the effectiveness of finite aperture illuminations with respect to synthesizing this ideal radiation pattern. For illustration some classical illuminations are scrutinized from the viewpoint of this new criterion.