Annular Walsh filters are derived from the rotationally symmetric annular Walsh functions which form a complete set of orthogonal functions that take on values either +1 or −1 over the domain specified by the inner and outer radii of the annulus. The value of any annular Walsh function is taken as zero from the centre of the circular aperture to the inner radius of the annulus. The three values 0, +1, and −1 in an annular Walsh function can be realized in a corresponding annular Walsh filter by using transmission values of zero amplitude (i.e., an obscuration), unity amplitude and zero phase, and unity amplitude and phase, respectively. Not only the order of the Walsh filter but also the size of the inner radius of the annulus provides an additional degree of freedom in tailoring of point spread function by using these filters for pupil plane filtering in imaging systems. In this report, we present the far-field amplitude characteristics of some of these filters to underscore their potential for effective use in several demanding applications like high-resolution microscopy, optical data storage, microlithography, optical encryption, and optical micromanipulation.
Far-field Diffraction Properties of Annular Walsh Filters