OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • 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)

View Full Text Article

Enhanced HTML    Acrobat PDF (244 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

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

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)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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).
  3. D. Elliott, K. Rao, Fast Transforms (Academic, Orlando, 1982), Chaps. 4 and 5.
  4. A. Siegman, “Quasi-fast Hankel transform,” Opt. Lett. 1, 13–15 (1977). [CrossRef]
  5. P. Murphy, N. Gallagher, “Fast algorithm for the computation of the zero-order Hankel transform,” J. Opt. Soc. Am. 73, 1130–1137 (1983). Contains references to other FFT-based Hankel transform algorithms. [CrossRef]
  6. G. Agrawal, M. Lax, “End correction in the quasi-fast Hankel transform for optical propagation problems,” Opt. Lett. 6, 171–173 (1981). [CrossRef] [PubMed]
  7. A. Oppenheim, G. Frisk, D. Martinez, “An algorithm for the numerical evaluation of the Hankel transform,” Proc. IEEE 66, 264–265 (1978). [CrossRef]
  8. S. Candel, “An algorithm for the Fourier–Bessel transform,” Comput. Phys. Commun. 23, 343–353 (1981). [CrossRef]
  9. V. Magni, V. Cerullo, S. Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,” J. Opt. Soc. Am. A 9, 2031–2033 (1992). [CrossRef]
  10. D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, J. Levy, “Optical diffraction of fractal figures: random Sierpinski carpets,” J. Phys. I (Paris) 1, 1433–1450 (1991).
  11. R. Barakat, E. Parshall, “Numerical evaluation of zero-order Hankel transforms using Filon quadrature philosophy,” Appl. Math. Lett. 9, 21–26 (1996). [CrossRef]
  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).
  14. L. Filon, “On a quadrature formula for trigonometric integrals,” Proc. R. Soc. Edin. 49, 38–47 (1928).
  15. C. Trantner, Integral Transforms in Mathematical Physics (Methuen, London, 1966), Chap. 6. This is the only book that contains full details of Filon’s work.
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  17. I. Gradshteyn, I. Ryzhk, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).
  18. I. Sneddon, Fourier Transforms (Dover, New York, 1995), Chap. 3.
  19. G. Watson, Theory of Bessel Functions (Cambridge, London, 1944).
  20. F. Oliver, ed., Royal Society Mathematical Tables: Vol. 7, Bessel Functions, Part III, Zeros and Associated Values (Cambridge University Press, Cambridge, 1960).
  21. R. Barakat, “Application of the sampling theorem to optical diffraction theory,” J. Opt. Soc. Am. 54, 920–930 (1964). [CrossRef]
  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited