Method to calculate lateral and axial gain factors of optical setups with a large solid angle

With the use of an approach that is equivalent to Heisenberg's uncertainty principle, it is possible to derive the axial and lateral gain factors of optical systems consisting of rotationally symmetric pupil plane filters with a large solid angle. We discuss the two- and three-dimensional cases. In contrast to previously published approaches, our results are valid for solid angles as large as 4 pi. The annular circular aperture is discussed as an example. Resolution gains for annular binary filters consisting of s transmitting rings are calculated, allowing the discussion of the optical properties of all rotationally symmetric binary filters in terms of gain factors. Optical systems with binary two-ring apertures are discussed; a special case is the 4Pi setup. Our novel method yields simple analytical formulas that are functions of the various aperture parameters. (C) 1999 Optical Society of America [S0740-3232(99)01411-8].