Multidimensional Fourier transforms, Parseval’s theorem, and Schwarz’s inequality, used in concert, show how to apodize a lens to minimize the nth moment of the irradiance in the diffraction pattern for a given central irradiance. Replacing Schwarz’s inequality with the calculus of variations, we can find the pupils that minimize quotients of certain different moments. In particular, the mean-square miss distance of a photon gun is minimized if the waves it launches have amplitudes that decrease as the Jo Bessel function moves from the center of the projecting lens and reach zero at the rim.
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