Bounds on the intensity and the variation of the far field F(u,v) of a given object ƒ(x,y) are developed in terms of the energy and the area of the object. An amplitude function ƒ(x,y) is determined for maximizing F(u0v0). The results are extended to objects with circular symmetry. The analysis is applied to the following apodization problem: Given a pupil of specified boundary R, a transmission function ƒ(x,y) of energy E is sought such that the energy of its far field in a region S of the u-v plane is maximum.
A. PAPOULIS, "Maximum Intensity of Diffraction Patterns and Apodization," J. Opt. Soc. Am. 57, 362-366 (1967)