OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 48, Iss. 22 — Aug. 1, 2009
  • pp: 4310–4319

Fast computation algorithm for the Rayleigh–Sommerfeld diffraction formula using a type of scaled convolution

Victor Nascov and Petre Cătălin Logofătu  »View Author Affiliations


Applied Optics, Vol. 48, Issue 22, pp. 4310-4319 (2009)
http://dx.doi.org/10.1364/AO.48.004310


View Full Text Article

Enhanced HTML    Acrobat PDF (751 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We describe a fast computational algorithm able to evaluate the Rayleigh–Sommerfeld diffraction formula, based on a special formulation of the convolution theorem and the fast Fourier transform. What is new in our approach compared to other algorithms is the use of a more general type of convolution with a scale parameter, which allows for independent sampling intervals in the input and output computation windows. Comparison between the calculations made using our algorithm and direct numeric integration show a very good agreement, while the computation speed is increased by orders of magnitude.

© 2009 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(260.1960) Physical optics : Diffraction theory
(070.2025) Fourier optics and signal processing : Discrete optical signal processing
(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: April 22, 2009
Revised Manuscript: June 24, 2009
Manuscript Accepted: June 26, 2009
Published: July 21, 2009

Citation
Victor Nascov and Petre Cătălin Logofătu, "Fast computation algorithm for the Rayleigh-Sommerfeld diffraction formula using a type of scaled convolution," Appl. Opt. 48, 4310-4319 (2009)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-22-4310


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 3.
  2. J. W. Cooley and J. W. Tookey, “An algorithm for the machine computation of the complex Fourier series,” Math. Comput. 19, 297-301 (1965). [CrossRef]
  3. L. P. Yaroslavsky, “Discrete transforms, fast algorithms, and point spread functions of numerical reconstruction of digitally recorded holograms,” in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds. (Hindawi2007), pp. 93-141.
  4. J. A. C. Veerman, J. J. Rusch, and H. P. Urbach, “Calculation of the Rayleigh-Sommerfeld diffraction integral by exact integration of the fast oscillating factor,” J. Opt. Soc. Am. A 22, 636-646 (2005). [CrossRef]
  5. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102-1110(2006). [CrossRef] [PubMed]
  6. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979). [CrossRef]
  7. N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A 15, 857-867 (1998). [CrossRef]
  8. L. B. Klebanov and J. F. Crouzet, “Quasi-convolutions and applications to coded images,” J. Math. Sci. 99, 1120-1126(2000). [CrossRef]
  9. D. Mustard, “Fractional convolution,” J. Austral. Math. Soci. Ser. B 40, 257-265 (1998). [CrossRef]
  10. P. C. Logofătu and D. Apostol, “The Fourier transform in optics: from continuous to discrete or from analogous experiment to digital calculus,” J. Optoelectron. Adv. Mat. 9, 2838-2846(2007).
  11. M. Mandal and A. Asif, Continuous and Discrete Time Signals and Systems (Cambridge U. Press, 2007).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 2.
  13. “An introduction to the sampling theorem,” National Semiconductor Application Note 236 (1980), http://www.national.com/an/AN/AN-236.pdf.
  14. D. H. Bailey and P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389-404(1991). [CrossRef]
  15. A. Bultheel and H. Martinez, “Computation of the fractional Fourier transform,” Appl. Comput. Harmon. Anal. 16, 182-202 (2004). [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.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited