OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 22, Iss. 4 — Apr. 1, 2005
  • pp: 636–646

Calculation of the Rayleigh–Sommerfeld diffraction integral by exact integration of the fast oscillating factor

Jan A. C. Veerman, Jurgen J. Rusch, and H. Paul Urbach  »View Author Affiliations

JOSA A, Vol. 22, Issue 4, pp. 636-646 (2005)

View Full Text Article

Enhanced HTML    Acrobat PDF (242 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We describe a numerical method that can be used to calculate the propagation of light in a medium of constant (possibly complex) index of refraction n. The method integrates the Rayleigh–Sommerfeld diffraction integral numerically. After an appropriate change of integration variables, the integrand of the diffraction integral is split into a slowly varying and an (often fast) oscillating quadratic factor. The slowly varying factor is approximated by a spline fit, and the resulting Fresnel integrals are subsequently integrated exactly. Although the method is not as fast as methods involving a fast Fourier transform, such as plane-wave propagation or Fresnel approximation, it is accurate over a greater range than these methods.

© 2005 Optical Society of America

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(260.0260) Physical optics : Physical optics

Original Manuscript: August 6, 2004
Manuscript Accepted: September 11, 2004
Published: April 1, 2005

Jan A. C. Veerman, Jurgen J. Rusch, and H. Paul 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)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986).
  2. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986). [CrossRef]
  3. P. Török, S. J. Hewlett, P. Varga, “On the series expansion of high-aperture, vectorial diffraction integrals,” J. Mod. Opt. 44, 493–503 (1997). [CrossRef]
  4. C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997). [CrossRef]
  5. S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001). [CrossRef]
  6. P. L. M. Put, H. P. Urbach, R. D. Morton, J. J. Rusch, “Resolution limit of optical disc mastering,” Jpn. J. Appl. Phys. 36, 539–548 (1997). [CrossRef]
  7. J. M. Brok, H. P. Urbach, “Rigorous model of the scat-tering of a focused spot by a grating and its application in optical recording,” J. Opt. Soc. Am. A 20, 256–272 (2003). [CrossRef]
  8. T. Gravelsaeter, J. J. Stamnes, “Diffraction by circular apertures. 1: Method of linear phase and amplitude approximation,” Appl. Opt. 21, 3644–3651 (1982). [CrossRef] [PubMed]
  9. J. J. Stamnes, “Hybrid integration technique for efficient and accurate computation of diffraction integrals,” J. Opt. Soc. Am. A 6, 1330–1342 (1989). [CrossRef]
  10. L. d’Arcio, J. J. M. Braat, H. J. Frankema, “Numerical evaluation of diffraction integrals for apertures of complicated shape,” J. Opt. Soc. Am. A 11, 2664–2674 (1994). [CrossRef]
  11. See, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  12. J. A. C. Veerman, “Evaluation of spot propagation calculations in 1D,” Internal Note, 2001.
  13. J. Stoer, Einführung in die Numerische Mathematik (Springer Verlag, Berlin, 1979).
  14. See http://www.tgs.com .

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