Far-field diffraction properties of radial Walsh filters
01 Jun 1986-Journal of The Optical Society of America A-optics Image Science and Vision (Optical Society of America)-Vol. 3, Iss: 6, pp 843-846
TL;DR: In this paper, a set of orthogonal radial filters for the pupil of an imaging system is presented, with analytical expressions for members of the set and their Hankel transforms of order zero.
Abstract: Radial Walsh functions form a closed set of orthogonal functions over a given finite interval, each function taking merely one constant value (either +1 or −1) in each of a finite number of subintervals into which the entire interval is divided. This set provides a remarkable set of orthogonal radial filters for the pupil of an imaging system. We report analytical expressions for members of the set and their Hankel transforms of order zero. Far-field diffraction characteristics, namely, the far-field amplitude distribution and the optical transfer functions, are presented for the first eight members of the set.
Citations
More filters
TL;DR: In this paper, a simple comprehensive treatment on the use of free-form optical elements, and of nonuniform optical windows, either for increasing focal depth or for tuning the depth of field, by controlling the influence of focus error on the modulation transfer function.
Abstract: We present a simple comprehensive treatment on the use of free-form optical elements, and of nonuniform optical windows, either for increasing focal depth [by regulating the width of the axial point spread function (PSF)] or for tuning the depth of field [by controlling the influence of focus error on the modulation transfer function (MTF)]. We employ the rising notation of pupil engineering, which incorporates techniques for controlling the spread of the axial PSF, as well as methods for governing the impact of focus errors on the MTF. Our discussion also includes the use of vortex lenses for designing nonconventional optical systems.
34 citations
TL;DR: Different types of Walsh functions in one and two dimensions are presented to demonstrate that a large class of pupil filters can be synthesized from them to cater to the various needs of diffraction pattern for tailoring transverse and/or axial resolution in microscopic applications.
29 citations
TL;DR: Observations on self-similarity in radial Walsh filters of various orders and corresponding axial intensity distributions provide valuable clues in tackling the inverse problem of synthesis of phase filter in accordance with prespecified axialintensity distributions.
Abstract: 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.
21 citations
TL;DR: The purpose of this work was to develop formulas and accurate numerical techniques for computation of the optical transfer function (OTF) in the general case of unrestricted aberration and with the following different forms of nonuniform real amplitude.
Abstract: The purpose of this work was to develop formulas and accurate numerical techniques for computation of the optical transfer function (OTF) in the general case of unrestricted aberration and with the following different forms of nonuniform real amplitude: (1) when the real amplitude is described by a polynomial, (2) a Gaussian distribution of real amplitude, and (3) a pupil with a central obstruction. The resulting computer program has been carefully tested and used to study the influence of nonuniform amplitude on the OTF in typical cases, for which detailed numerical results are given.
19 citations
TL;DR: In this article, the authors derived the rotationally symmetric annular Walsh filters from the Walsh functions, which form a complete set of orthogonal functions that take on values either
Abstract: 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
15 citations
Additional excerpts
...For systems with rotational symmetry about the axis, radial Walsh functions [16] have been developed as a special case ofWalsh functions in polar co-ordinates and they were proved useful in the treatment of apodization problems [17, 18]....
[...]
References
More filters
TL;DR: In this article, the authors studied a new closed set of functions normal and orthogonal on the interval (0, 1) for the interval 0 5 x 5 1, where each function takes only the values + 1 and − 1, except at a finite number of points of discontinuity, where it takes the value zero.
Abstract: A set of normal orthogonal functions {χ} for the interval 0 5 x 5 1 has been constructed by Haar†, each function taking merely one constant value in each of a finite number of sub-intervals into which the entire interval (0, 1) is divided. Haar’s set is, however, merely one of an infinity of sets which can be constructed of functions of this same character. It is the object of the present paper to study a certain new closed set of functions {φ} normal and orthogonal on the interval (0, 1); each function φ has this same property of being constant over each of a finite number of sub-intervals into which the interval (0, 1) is divided. In fact each function φ takes only the values +1 and −1, except at a finite number of points of discontinuity, where it takes the value zero. The chief interest of the set φ lies in its similarity to the usual (e.g., sine, cosine, Sturm-Liouville, Legendre) set of orthogonal functions, while the chief interest of the set χ lies in its dissimilarity to these ordinary sets. The set φ shares with the familiar sets the following properties, none of which is possessed by the set χ: the nth function has n−1 zeroes (or better, sign-changes) interior to the interval considered, each function is either odd or even with respect to the mid-point of the interval, no function vanishes identically on any sub-interval of the original interval, and the entire set is uniformly bounded. Each function χ can be expressed as a linear combination of a finite number of functions φ, so the paper illustrates the changes in properties which may arise from a simple orthogonal transformation of a set of functions. In § 1 we define the set χ and give some of its principal properties. In § 2 we define the set φ and compare it with the set χ. In § 3 and § 4 we develop some of the properties of the set φ, and prove in particular that every continuous function of bounded variation can be expanded in terms of the φ’s and that every continuous function can be so developed in the sense not of convergence of the series but of summability by the first Cesaro mean. In § 5 it is proved that there exists a continuous function which cannot be
918 citations
TL;DR: In this paper, a theoretical study of the possibilities of using an annular aperture to increase the focal depth of a photographic objective is made and it is shown that for a given gain in focal depth the loss in speed is the same for both annular apertures and conventional stopping down.
Abstract: A theoretical study is made of the possibilities of using an annular aperture to increase the focal depth of a photographic objective. It is shown that for a given gain in focal depth the loss in speed is the same for both annular apertures and conventional stopping down. For images of isolated point objects, the definition is improved by using an annular stop. The gain in focal depth is less for off-axis points, but it is found that, for example, a factor of 2.7 in focal depth gained by means of an annular aperture is barely affected at a field angle of 30°.
294 citations
TL;DR: To improve the imaging properties of a defocused optical system, the use of shaded apertures is studied theoretically and experimentally and shows that near focus the OTF for T(A) has higher values in the low frequency region than has either T(B) or T(c).
Abstract: To improve the imaging properties of a defocused optical system, the use of shaded apertures is studied theoretically and experimentally. The study is based on the optical transfer function (OTF). The two shaded apertures studied are the type in which the amplitude transmittance decreases gradually from the center of the pupil toward its rim, T(A), and the type in which the amplitude transmittance decreases from its rim toward the center, T(B). For comparison, the effects achieved with a clear aperture, T(C), are included. The results of the calculations show that near focus the OTF for T(A) has higher values in the low frequency region than has either T(B) or T(c). When the system is defocused, the shaded aperture of the type T(A) yields an improved defocused image that is faithful to the outline of the object. The quality of the defocused image obtained with T(B) is worsened. When the OTF is used as a means for judging the quality of the defocused image, the two necessary conditions on the functions appear to be that the OTF (1) must be a monotonically decreasing function and (2) must be nonnegative. These conditions are confirmed by experiment. Since the transmittance variation of the shaded apertures is achieved by absorption, the effects due to the resultant decreases in light level are also considered.
171 citations