Multidimensional Fourier transforms, Parseval’s theorem, and Schwarz’s inequality, used in concert, show how to apodize a lens to minimize the <i>n</i>th 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 <i>J</i><sub>o</sub> Bessel function moves from the center of the projecting lens and reach zero at the rim.
© 1969 Optical Society of America
C. W. MCCUTCHEN, "Two Families of Apodization Problems," J. Opt. Soc. Am. 59, 1163-1166 (1969)