Antenna beam patterns which retain shape with defocusing

A narrow beam is formed by radiation from a large aperture having a flat phase front. Any curvature of the phase front, as by defocusing, causes widening of the beam and reduction of the directive gain. Certain symmetric and antisymmetric forms of aperture excitation, based on the probability function, have been known to yield radiation patterns of the same shape. It is found that these patterns retain their shape when widened by defocusing, so that the effects of defocusing can be expressed by simple and exact formulas. Also they retain some ideal relations that are peculiar to their use as sum and difference patterns in a tracking radar of the simultaneous-lobing or monopulse type. The simplification results from the fact that the attenuation for tapering the aperture excitation, and the phase shift by defocusing, are both proportional to the square of the aperture coordinates from the center. The amount of defocusing is then expressed by the constant ratio of these two quantities in radians per napier. While these forms ideally require a line or circle aperture of unlimited extent, they can be approximated with an aperture of finite extent. Therefore the resulting simple concepts and rules are helpful in practical situations.