Modified Luneburg Apodization Problem I

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Abstract
One of the modified Luneburg apodization problems suggested by Wilkins as an unsolved apodization problem is solved in this paper. The problem is to find the pupil function which maximizes the total flux inside a prespecified circle in the Fraunhofer diffraction image plane, subject to the conditions that the pupil function satisfies the inequality |T(r)|≤1 and that the total flux passing the aperture is specified to have a certain value. The method used is the calculus of variations leading to a nonlinear integral equation which is solved by successive approximations using the dummy functions. The resultant pupil functions obtained are compared with those obtained by Straubel and Lansraux and Boivin.

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