Bounds on the intensity and the variation of the far field <i>F</i>(<i>u,v</i>) of a given object ƒ(<i>x,y</i>) are developed in terms of the energy and the area of the object. An amplitude function ƒ(<i>x,y</i>) is determined for maximizing <i>F</i>(<i>u</i><sub>0</sub><i>v</i><sub>0</sub>). The results are extended to objects with circular symmetry. The analysis is applied to the following apodization problem: Given a pupil of specified boundary <i>R</i>, a transmission function ƒ(<i>x,y</i>) of energy <i>E</i> is sought such that the energy of its far field in a region <i>S</i> of the <i>u</i>-<i>v</i> plane is maximum.
A. PAPOULIS, "Maximum Intensity of Diffraction Patterns and Apodization," J. Opt. Soc. Am. 57, 362-366 (1967)