Expressions are obtained for the three-dimensional intensity distribution in the error-free diffraction patterns of semitransparent and phase annuli. The results are discussed and illustrated graphically. Annular apertures have been used for improving the resolving power and focal depth of optical instruments; the most useful situation for a given central-obstruction ratio is where the central obstruction is completely opaque. However, by use of a phase annulus of the same dimension, with a phase step of π for the central region, some further improvement in the resolving power and focal depth is obtained for central-obstruction ratios up to 0.5. This improvement is illustrated experimentally.
BRIAN J. THOMPSON, "Diffraction by Semitransparent and Phase Annuli," J. Opt. Soc. Am. 55, 145-148 (1965)