Apodization for maximum encircled energy with specified over-all transmittance

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The Luneberg's third apodization problem is solved by taking into account the passivity of an optical system. The problem is to find the optimum pupil function such that the total flux in a circle of specified radius (encircled energy) is made a maximum for prespecified over-all transmittance. The problem is treated for a rotationally symmetric optical system without aberrations. The method used to solve this problem is the calculus of variations of unconventional type involving the constraint of inequalities due to the passivity of the optical system. The solution is given in algebraic form and numerically evaluated. Discussion is finally given for numerical results of the optimum pupil function and corresponding diffraction pattern.

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