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. 1. Introduction Annular apertures and different types of ring-shaped apertures continue to be investigated for catering to the growing exigencies in diverse applications, for example, high resolution microscopy, optical data storage, microlithography, optical encryption, and optical micromanipulation [1–5]. Not only for obvious energy considerations but also for their higher inherent potential in delivering complex far-field amplitude distributions, annular phase filters are being investigated in different contexts [6–9]. A systematic study on the use of phase filters on annular pupils can be conveniently carried out with the help of annular Walsh filters derived from the annular Walsh functions. Walsh functions form a closed set of normal orthogonal functions over a given finite interval and take on values +1 or ？1, except at a finite number of points of discontinuity, where they take the value zero [10, 11]. The order of a Walsh function is directly related to the number of its zero crossings or phase transitions within the specified domain, and they constitute a closed set of normal orthogonal functions over the specified interval. They have the interesting property that an approximation of a continuous function over a finite interval by a finite set of Walsh functions leads to a piecewise constant approximation to the function. Walsh filters of various orders may be obtained from
References
[1]
G. T. di Francia, “Nuovo pupille superrisolvente,” Atti della Fondazione Giorgio Ronchi, vol. 7, pp. 366–372, 1952.
[2]
E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proceedings of the Physical Society B, vol. 66, no. 2, pp. 145–149, 1953.
[3]
A. Boivin, Théorie et Calcul des Figures de Diffraction de Revolution, Gauthier-Villars, Paris, France, 1964.
[4]
Z. S. Hegedus, “Annular pupil arrays—application to confocal scanning,” Optica Acta, vol. 32, pp. 815–826, 1985.
[5]
C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Applied Optics, vol. 43, no. 22, pp. 4322–4327, 2004.
[6]
T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” Journal of the Optical Society of America A, vol. 14, no. 7, pp. 1637–1646, 1997.
[7]
H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Applied Optics, vol. 43, no. 34, pp. 6242–6247, 2004.
[8]
M. Yun, L. Liu, J. Sun, and D. Liu, “Three-dimensional superresolution by three-zone complex pupil filters,” Journal of the Optical Society of America A, vol. 22, no. 2, pp. 272–277, 2005.
[9]
M. Yun, M. Wang, and L. Liu, “Super-resolution with annular binary phase filter in the 4Pi-confocal system,” Journal of Optics A, vol. 7, no. 11, pp. 640–644, 2005.
[10]
J. L. Walsh, “A closed set of normal orthogonal functions,” The American Journal of Mathematics, vol. 55, pp. 5–24, 1923.
[11]
K. G. Beauchamp, Walsh Functions and Their Applications, Academic, New York, NY, USA, 1985.
[12]
C. J. R. Sheppard, “Binary phase filters with a maximally-flat response,” Optics Letters, vol. 36, no. 8, pp. 1386–1388, 2011.
[13]
H. F. Harmuth, Transmission of Information by Orthogonal Functions, Springer, Berlin, Germany, 1972.
[14]
H. C. Andrews, Computer Techniques in Image Processing, Academic, New York, NY, USA, 1970.
[15]
M. De and L. N. Hazra, “Walsh functions in problems of optical imagery,” Optica Acta, vol. 24, pp. 221–234, 1977.
[16]
L. N. Hazra and A. Guha, “Farfield diffraction properties of radial Walsh filters,” Journal of the Optical Society of America A, vol. 3, pp. 843–846, 1986.
[17]
L. N. Hazra and A. Banerjee, “Application of Walsh function in generation of optimum apodizers,” Journal of Optics, vol. 5, pp. 19–26, 1976.
[18]
L. N. Hazra, “A new class of optimum amplitude filters,” Optics Communications, vol. 21, no. 2, pp. 232–236, 1977.
[19]
L. N. Hazra, “Walsh filters for tailoring of resolution in microscopic imaging,” Micron, vol. 38, no. 2, pp. 129–135, 2007.
[20]
M. Born and E. Wolf, Principles of Optics, Pergamon, Oxford, UK, 1980.
[21]
H. H. Hopkins, “Canonical and real space coordinates used in the theory of image formation,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, Eds., vol. 9, p. 307, Academic, New York, NY, USA, 1983.
[22]
E. Fernández, A. Márquez, S. Gallego, R. Fuentes, C. García, and I. Pascua, “Hybrid ternary modulation applied to multiplexing holograms in photopolymers for data page storage,” Journal of Lightwave Technology, vol. 28, pp. 776–783, 2010.
[23]
A. Au, C.-S. Wu, S.-T. Wu, and U. Efron, “Ternary phase and amplitude modulations using a twisted nematic liquid crystal spatial light modulator,” Applied Optics, vol. 34, no. 2, pp. 281–284, 1995.
[24]
S. Mukhopadhyay, S. Sarkar, K. Bhattacharya, and L. N. Hazra, “Polarisation phase shifting interferometric technique for phase calibration of a reflective phase spatial light modulator,” Optical Engineering, vol. 52, Article ID 035602, 2 pages, 2013.
[25]
R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications,” Proceedings of the IEEE, vol. 90, no. 2, pp. 222–244, 2002.
