Apodization for minimizing the second moment of the intensity distribution in the Fraunhofer diffraction pattern (I)

A detailed solution is given to the apodization problem for finding the pupil function (amplitude distribution over the exit pupil) which minimizes the second moment of the intensity distribution in the Fraunhofer diffraction pattern on the condition that the total energy passing through the aperture be constant. The problem is in the context of Fraunhofer diffraction pattern, and aberration-free systems with both circular and slit apertures are studied. The method used to solve this problem is the calculus of variations leading to the Euler-Lagrange differential equation for the desired pupil functions. Discussion on the pupil functions obtained is conducted.