Apodization for systems of high f-number

The exit pupil distribution that maximizes the mean encircled energy fraction over some interval along a system’s line of symmetry is found here for both two-and three-dimensional geometries. Such a merit function has been optimized previously by using the Debye model for the diffracted field, but this is not valid for the systems studied here; the more general Kirchhoff integral must be used. Optimal pupil functions for these systems are shown to depend on an additional parameter and represent a new family of solutions with nontrivial phase. Taylor series are derived for these functions, and they provide a powerful verification of the numerical results.