OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 45, Iss. 6 — Feb. 20, 2006
  • pp: 1102–1110

Fast-Fourier-transform based numerical integration method for the Rayleigh–Sommerfeld diffraction formula

Fabin Shen and Anbo Wang  »View Author Affiliations

Applied Optics, Vol. 45, Issue 6, pp. 1102-1110 (2006)

View Full Text Article

Enhanced HTML    Acrobat PDF (1206 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The numerical calculation of the Rayleigh–Sommerfeld diffraction integral is investigated. The implementation of a fast-Fourier-transform (FFT) based direct integration (FFT-DI) method is presented, and Simpson's rule is used to improve the calculation accuracy. The sampling interval, the size of the computation window, and their influence on numerical accuracy and on computational complexity are discussed for the FFT-DI and the FFT-based angular spectrum (FFT-AS) methods. The performance of the FFT-DI method is verified by numerical simulation and compared with that of the FFT-AS method.

© 2006 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(050.1960) Diffraction and gratings : Diffraction theory
(350.5500) Other areas of optics : Propagation

ToC Category:
Diffraction and Gratings

Original Manuscript: March 15, 2005
Revised Manuscript: September 9, 2005
Manuscript Accepted: September 22, 2005

Fabin Shen and Anbo Wang, "Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula," Appl. Opt. 45, 1102-1110 (2006)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 8.
  2. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987), Chap. 10.
  3. N. Mukunda, "Consistency of Rayleigh's diffraction formulas with Kirchhoff's boundary conditions," J. Opt. Soc. Am. 52, 336-337 (1962). [CrossRef]
  4. E. Wolf and E. W. Marchand, "Comparison of the Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction at an aperture," J. Opt. Soc. Am. 54, 587-594 (1964). [CrossRef]
  5. J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974-980 (1979). [CrossRef]
  6. J. C. Heurtley, "Scalar Rayleigh-Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points," J. Opt. Soc. Am. 63, 1003-1008 (1973). [CrossRef]
  7. M. Totzeck, "Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects," J. Opt. Soc. Am. A 8, 27-32 (1991). [CrossRef]
  8. A. M. Steane and H. N. Rutt, "Diffractions in the near field and the validity of the Fresnel approximation," J. Opt. Soc. Am. A 6, 1809-1814 (1989). [CrossRef]
  9. W. H. Southwell, "Validity of the Fresnel approximation in the near field," J. Opt. Soc. Am. 71, 7-14 (1981). [CrossRef]
  10. E. Lalor, "Conditions for the validity of the angular spectrum of plane waves," J. Opt. Soc. Am. 58, 1235-1237 (1968). [CrossRef]
  11. J. J. Stamnes, "Focusing of two dimensional waves," J. Opt. Soc. Am. 71, 15-20 (1981). [CrossRef]
  12. 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]
  13. J. A. Hudson, "Fresnel-Kirchhoff diffraction in optical systems: an approximate computational algorithm," Appl. Opt. 23, 2292-2295 (1984). [CrossRef] [PubMed]
  14. C. Kopp and P. Meyrueis, "Near-field Fresnel diffraction: improvement of a numerical propagator," Opt. Commun. 158, 7-10 (1998). [CrossRef]
  15. J. Li, Z. Fan, and Y. Fu, "The FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality," in Lasers in Material Processing and Manufacturing, S. Deng, T. Okada, K. Behler, and X. Wang, eds., Proc. SPIE 4915, 180-186 (2002). [CrossRef]
  16. N. Delen and B. Hooker, "Verification and comparison of a fast Fourier transform-based full diffraction method for tilted and offset planes," Appl. Opt. 40, 3525-2531 (2001). [CrossRef]
  17. C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford U. Press, 1998), Chap. 7.
  18. W. L. Briggs and V. E. Henson, The DFT: An Owner's Manual for the Discrete Fourier Transform (Society for Industrial and Applied Mathematics, 1995). [CrossRef]
  19. H. Osterberg and L. W. Smith, "Closed solutions of Rayleigh's integral for axial points," J. Opt. Soc. Am. 51, 1050-1054 (1961). [CrossRef]
  20. A. Dubra and J. A. Ferrari, "Diffracted field by an arbitrary aperture," Am. J. Phys. 67, 87-92 (1999). [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