Abstract

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 J_o Bessel function moves from the center of the projecting lens and reach zero at the rim.

© 1969 Optical Society of America

Analytic Solution of Two Apodization Problems

David Slepian
J. Opt. Soc. Am. 55(9) 1110-1115 (1965)

Solution of the Luneberg Apodization Problems*

Richard Barakat
J. Opt. Soc. Am. 52(3) 264-275 (1962)

Luneberg Apodization Problems

J. Ernest Wilkins
J. Opt. Soc. Am. 53(4) 420-424 (1963)

Diffracted Wave Fields Expressible by Plane-Wave Expansions Containing only Homogeneous Waves*

George C. Sherman
J. Opt. Soc. Am. 59(6) 697-711 (1969)

Basic Duality in the Algebraic Formulation of Electromagnetic Diffraction

W. Duane Montgomery
J. Opt. Soc. Am. 59(7) 804-811 (1969)