Walsh filters for tailoring of resolution in microscopic imaging

Extended axial irradiances: Barker rings

For extending focal depth we employ a set of transparent concentric rings, which are coded with the Barker sequences of length L. At the neighborhood of the paraxial focal plane, these transparent masks generate an axial uniform distribution, which is modulated with sinusoidal variations. For imaging applications, one can extend focal depth if the Barker length is congruent to unity modulo 4. And, for optical trapping, a bottle neck irradiance distribution is generated if the Barker length is congruent to three modulo 4.

Reducing field depth: annular Hadamard masks

For optically narrowing field depth, we explore the use of annular masks that are coded with the elements of Hadamard matrices. We show that with no further reduction of light throughput, these annular masks can enhance the influence of focus error on the modulation transfer functions (MTF). We report numerical evaluations of the irradiance point spread functions (PSFs), their associated MTFs, and of some digitally generated images. For complementing our assessments, we evaluate the Fisher information of these Walsh-Hadamard, annular masks. We suggest an optical technique for breaking 3-D scenes into several 2-D sliced frames by translating out-of-focus images into background noise.

Multiple-plane image formation by Walsh zone plates

A radial Walsh filter is a phase binary diffractive optical element characterized by a set of concentric rings that take the phase values 0 or π, corresponding to the values + 1 or -1 of a given radial Walsh function. Therefore, a Walsh filter can be re-interpreted as an aperiodic multifocal zone plate, capable to produce images of multiple planes simultaneously in a single output plane of an image forming system. In this paper, we experimentally demonstrate for the first time the focusing capabilities of these structures. Additionally, we report the first achievement of images of multiple-plane objects in a single image plane with these aperiodic diffractive lenses.

Self-similarity in radial Walsh filters and axial intensity distributions in the far-field diffraction pattern

Pupil plane filtering by radial Walsh filters is a convenient technique for tailoring the axial intensity distribution near the focal plane of a rotationally symmetric imaging system. Radial Walsh filters, derived from radial Walsh functions, form a set of orthogonal phase filters that take on values either 0 or π phase, corresponding to +1 or -1 values of the radial Walsh functions over prespecified annular regions of the circular filter. Order of these filters is given by the number of zero-crossings, or equivalently phase transitions within the domain over which the set is defined. In general, radial Walsh filters are binary phase zone plates, each of them demonstrating distinct focusing characteristics. The set of radial Walsh filters can be classified into distinct groups, where the members of each group possess self-similar structures. Self-similarity can also be observed in the corresponding axial intensity distributions. These observations provide valuable clues in tackling the inverse problem of synthesis of phase filter in accordance with prespecified axial intensity distributions. This paper reports our observations on self-similarity in radial Walsh filters of various orders and corresponding axial intensity distributions.

Toraldo filters with concentric unequal annuli of fixed phase by evolutionary programming

The resolving power of an optical imaging system is limited by residual aberrations and diffraction effects. The Rayleigh-Abbe diffraction limit of resolution corresponds to radius of the central lobe of the point spread function of an aberration free diffraction limited system. An attempt to circumvent this limitation was proposed by Toraldo di Francia, who showed that suitable pupil plane filtering can overcome this resolution limit, albeit over a restricted field. This paper reports results of our investigations on the use of evolutionary programming to obtain globally or quasi-globally optimum solutions in synthesis of lossless Toraldo filters consisting of concentric unequal area zones of fixed phase.