Detailed solutions are given to the Luneberg apodization problems set by Luneberg. All problems are in the context of Fraunhofer diffraction and only aberration-free systems are studied. The first problem is to determine the pupil function (amplitude distribution over the exit pupil) which maximizes the central illuminance subject to the condition that the total energy passing through the aperture be constant. The second problem is to determine the pupil function which maximizes the central illuminance while decreasing the radius of the first zero of the diffraction pattern a prespecified amount. The third problem is to find a pupil function which maximizes the total flux through a prespecified circle (for a circular aperture) or narrow rectangle (for a slit aperture). The methods used are the calculus of variations leading to Fredholm integral equations for the desired pupil functions. Both circular and slit apertures are studied.

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