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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 15, Iss. 3 — Mar. 1, 1998
  • pp: 652–659

Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation

Richard Barakat, Elaine Parshall, and Barbara H. Sandler  »View Author Affiliations


JOSA A, Vol. 15, Issue 3, pp. 652-659 (1998)
http://dx.doi.org/10.1364/JOSAA.15.000652


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Abstract

Our purpose is to bring to the attention of the optical community our recent work on the numerical evaluation of zero-order Hankel transforms; such techniques have direct application in optical diffraction theory and in optical beam propagation. The two algorithms we discuss (Filon–Simpson and Filon-trapezoidal) are reasonably fast and very accurate; furthermore, the errors incurred are essentially independent of the magnitude of the independent variable. Both algorithms are then compared with the recent (fast-Fourier-transform-based Hankel transform algorithm developed by Magni, Cerullo, and Silvestri (MCS algorithm) [J. Opt. Soc. Am. A 9, 2031 (1992)] and are shown to be superior. The basic assumption of these algorithms is that the term in the integrand multiplying the Bessel function is relatively smooth compared with the oscillations of the Bessel function. This condition is violated when the inverse Hankel transform has to be computed, and the Filon scheme requires a very large number of quadrature points to achieve even moderate accuracy. To overcome this deficiency, we employ the sampling expansion (Whittaker’s cardinal function) to evaluate numerically the inverse Hankel transform.

© 1998 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(050.1960) Diffraction and gratings : Diffraction theory
(350.5500) Other areas of optics : Propagation

History
Original Manuscript: February 20, 1997
Revised Manuscript: September 15, 1997
Manuscript Accepted: September 22, 1997
Published: March 1, 1998

Citation
Richard Barakat, Elaine Parshall, and Barbara H. Sandler, "Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation," J. Opt. Soc. Am. A 15, 652-659 (1998)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-3-652


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References

  1. R. Barakat, “The numerical evaluation of diffraction integrals,” in The Computer in Optical Research, R. Frieden, ed. (Springer, New York, 1980), Chap. 2.
  2. L. Bingham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N. J., 1988).
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  12. R. Barakat, B. Sandler, “Filon trapezoidal schemes for Hankel transforms of orders zero and one,” Appl. Math. Lett. (to be published).
  13. R. Barakat, B. Sandler, “Numerical evaluation for first-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. (to be published).
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  22. R. Barakat, “Solution to an Abel integral equation for bandlimited functions by means of sampling theorems,” J. Math. Phys. (Cambridge, Mass.) 43, 332–335 (1964).
  23. A. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977). [CrossRef]

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