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Applied Optics

Applied Optics


  • Editor: Glenn D. Boreman
  • Vol. 44, Iss. 36 — Dec. 20, 2005
  • pp: 7677–7696

Far-field diffraction patterns of circular sectors and related apertures

Gonzalo Urcid and Alfonso Padilla  »View Author Affiliations

Applied Optics, Vol. 44, Issue 36, pp. 7677-7696 (2005)

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In studies of scalar diffraction theory and experimental practice, the basic geometric shape of a circle is widely used as an aperture. Its Fraunhofer diffraction pattern has a simple mathematical expression easily determined by use of the Fourier–Bessel transform. However, it may require considerable mathematical effort to determine the far-field diffraction patterns of aperture shapes related to the circular geometry. From a computational point of view, the mathematical difficulties posed by other aperture geometries as well as more-general apertures with irregular shapes can be surpassed by means of optical setups or fast numerical algorithms. Unfortunately, no additional insight or information can be obtained from their exclusive application, as would be the case if mathematical formulas were available. The research presented here describes the far-field diffraction patterns of single-sector apertures as well as their extension to double symmetrical sectors and to sector wheels formed by interleaved transparent sectors of equal angular size; in each case, full or annular sectors are considered. The analytic solutions of their far-field amplitude distribution are given here in terms of a series of Bessel functions, some interesting properties are deduced from these solutions, and several examples are provided to illustrate graphically the results obtained from approximate numerical computations. Our results have been verified numerically with the fast-Fourier-transform algorithm and experimentally by means of a spherical wavefront–single-lens Fourier-transform architecture.

© 2005 Optical Society of America

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.1220) Diffraction and gratings : Apertures
(050.1940) Diffraction and gratings : Diffraction
(050.1960) Diffraction and gratings : Diffraction theory
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(070.2590) Fourier optics and signal processing : ABCD transforms

ToC Category:
Diffraction and Gratings

Original Manuscript: July 6, 2005
Manuscript Accepted: August 16, 2005
Published: December 20, 2005

Gonzalo Urcid and Alfonso Padilla, "Far-field diffraction patterns of circular sectors and related apertures," Appl. Opt. 44, 7677-7696 (2005)

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