A method of evaluating the two-dimensional Fourier transform of the point image intensity distribution for an annular aperture is presented. Since all three integrals involved are of the same basic form, it is shown that an analytic solution can be obtained by integrating the general expression with respect to a parameter. The result agrees with Steel’s expression obtained by convolving the aperture function with its complex conjugate. A family of transfer functions is then plotted with the ratio of inner to outer radius of the annulus as a parameter.
EDWARD L. O’NEILL, "Transfer Function for an Annular Aperture," J. Opt. Soc. Am. 46, 285-288 (1956)