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Proceedings ArticleDOI

Asymmetrical PSF by Azimuthal Walsh filters

TL;DR: In this article, the effects of phase asymmetry in the azimuthal Walsh filters on the PSF in the Far-Field diffraction pattern were investigated and the results showed that the inherent asymmetry of the filters leads to asymmetrical point spread function (PSF) in the far field diffraction patterns.
Abstract: Azimuthal Walsh Functions can be utilized to obtain an orthogonal set of Azimuthal Walsh Filters where the transmission values of +1 and −1 are achieved by using 0 and pi phase respectively. The inherent asymmetry in the azimuthal Walsh filters leads to asymmetrical Point Spread Function (PSF) in the far field diffraction pattern. This report presents results of our preliminary investigation on the effects of phase asymmetry in the filters on the PSF in the Far-field patterns.
Citations
More filters
Book ChapterDOI
01 Jan 2019
TL;DR: In this paper, the authors reported the possibility of modifying the beam structure around the far-field plane by diffraction characteristics of azimuthal Walsh filters placed on the exit pupil plane when computed analytically.
Abstract: Azimuthal Walsh filters, derived from radially invariant azimuthal Walsh functions, can be used as an effective tool for producing 2D and 3D light structures near the focal plane of a rotationally symmetric imaging system by manipulating the far-field diffraction characteristics when used as pupil filters. Starting with the definition of azimuthal Walsh functions and using the scalar diffraction theory, this research work reports the possibility of modifying the beam structure around the far-field plane by diffraction characteristics of azimuthal Walsh filters placed on the exit pupil plane when computed analytically. The asymmetrical beam produced due to the inherent phase asymmetries introduced by azimuthal Walsh filters may find many important applications in micro- and nano-photonics.

3 citations

Book ChapterDOI
01 Jan 2015
TL;DR: Binary polar Walsh filters derived from two dimensional polar Walsh functions provide an effective tool for tailoring the three dimensional intensity distributions of a point spread function (PSF) in the pupil plane in a diffraction limited imaging system.
Abstract: Binary polar Walsh filters derived from two dimensional polar Walsh functions provides an effective tool for tailoring the three dimensional intensity distributions of a point spread function (PSF) in the pupil plane in a diffraction limited imaging system. This paper reports the study of radial as well as azimuthally variant polar Walsh filters and observation on their imaging characteristics near the focal plane of a rotationally symmetric imaging system.

2 citations

Posted ContentDOI
14 Oct 2022
TL;DR: In this article , a Walsh function filter of fixed orders and annular obstruction was used to generate axially polarized multiple spots, transversely polarized multiple holes,transversely polarized multi spots, and axially and transversely polarised multi holes with extended central annular region.
Abstract: Abstract Axially splitted multi foci(AMF) are numerically generated by tight focusing of higher order cylindrical vector beam(HCVB) through an annular Walsh filter .Here we report that by properly manipulating the parameters of HCVB such as initial phase( φ 0 ),topological charge ( m ), polarization rotation angle or azimuthal index ( a ) and suitably phase modulated with a Walsh function filter of fixed orders ( n ) and annular obstruction (ε) ,onecan generate axially polarized multiple spots,transversely polarized multiple holes,transversely polarized multi spots,axially and transversely polarized multi holes with extended central annular region.These distributions may be useful for multiple optical trapping and axial superresolution microscopy.
References
More filters
Journal ArticleDOI
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


"Asymmetrical PSF by Azimuthal Walsh..." refers background in this paper

  • ...INTRODUCTION Walsh functions are defined as a complete set of normal orthogonal functions over a given finite interval and takes on the values +1 or -1[1,2] over a finite domain where the value becomes zero, and the number of zero crossings within specified domain gives the order of the Walsh Function....

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Book
01 Jan 1969
TL;DR: When you read more every page of this transmission of information by orthogonal functions, what you will obtain is something great.
Abstract: Read more and get great! That's what the book enPDFd transmission of information by orthogonal functions will give for every reader to read this book. This is an on-line book provided in this website. Even this book becomes a choice of someone to read, many in the world also loves it so much. As what we talk, when you read more every page of this transmission of information by orthogonal functions, what you will obtain is something great.

366 citations

Journal ArticleDOI
TL;DR: In this article, the form of the far-field diffraction pattern of an aperture bounded by arcs and radii of concentric circles whose dimensions are large compared with the wavelength of the incident radiant flux and over which the amplitude and phase are constant.
Abstract: Some equations are derived for computing the form of the far-field diffraction pattern of an aperture bounded by arcs and radii of concentric circles whose dimensions are large compared with the wavelength of the incident radiant flux and over which the amplitude and phase are constant. These equations are derived in such a way that it is possible not only to compute the form of the far-field diffraction pattern of individual apertures bounded by arcs and radii of concentric circles, but also for combinations of such apertures over which the amplitudes and phases are constant but differ from aperture to aperture. With the aid of an IBM 7090 computer, these equations have been used to compute the forms of the far-field diffraction patterns of a 60° sector aperture, a semicircular aperture, and an aperture formed by arcs and radii from concentric circles, and three-dimensional models have been constructed to describe the radiant flux distribution. A start has also been made on the more complicated problem of multiple apertures by computing the form of the far-field diffraction pattern of two adjacent semicircular apertures, over which the amplitudes and phases are constant in the two apertures, but the amplitudes in the two apertures have been assumed to be equal and the phases different. Some corresponding photographed far-field diffraction patterns are also included for comparison purposes and to illustrate the wide variety possible.

44 citations


"Asymmetrical PSF by Azimuthal Walsh..." refers methods in this paper

  • ...The far-field amplitude distribution due to a sector shaped aperture is: R rdrdr f rirtpA 0 2 0 )4.....()cos('2exp),(),( Following analysis of Mahan [6], Som and Lessard [7 ], the amplitude of PSF can be written as : )5(....................;,;,),( piSpCTpA where T is the transmission of the pupil function....

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  • ...[6] A. I. Mahan, C.V. Bitterli, and S.M. Cannon, “Far-field diffraction patterns of Single and Multiple Apertures bounded by arcs and radii of concentric circles, Journal of Optical Society of America, Volume 54, Number 6, June 1964....

    [...]

  • ...Following analysis of Mahan [6], Som and Lessard [7 ], the amplitude of PSF can be written as :...

    [...]

Journal ArticleDOI
TL;DR: In this article, a new class of optimum amplitude filters for the maximization of the factor of encircled energy within a pre-specified central core area of the far-field diffraction pattern is presented.

35 citations

Journal ArticleDOI
TL;DR: A reformulation of Walsh functions in polar coordinates facilitates the analysis which is presented, along with a restatement of the basic properties of these functions pertinent to the purpose.
Abstract: Walsh functions provide a viable and interesting tool in the treatment of problems of optical imagery. A reformulation of Walsh functions in polar coordinates facilitates the analysis which is presented, along with a restatement of the basic properties of these functions pertinent to the purpose. Some numerical results of aberrational diffraction images are cited to illustrate the compatibility of the proposed approach.

30 citations