OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 29, Iss. 7 — Jul. 1, 2012
  • pp: 1459–1469

Scalar diffraction field calculation from curved surfaces via Gaussian beam decomposition

Erdem Şahin and Levent Onural  »View Author Affiliations

JOSA A, Vol. 29, Issue 7, pp. 1459-1469 (2012)

View Full Text Article

Enhanced HTML    Acrobat PDF (745 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We introduce a local signal decomposition method for the analysis of three-dimensional (3D) diffraction fields involving curved surfaces. We decompose a given field on a two-dimensional curved surface into a sum of properly shifted and modulated Gaussian-shaped elementary signals. Then we write the 3D diffraction field as a sum of Gaussian beams, each of which corresponds to a modulated Gaussian window function on the curved surface. The Gaussian beams are propagated according to a derived approximate expression that is based on the Rayleigh–Sommerfeld diffraction model. We assume that the given curved surface is smooth enough that the Gaussian window functions on it can be treated as written on planar patches. For the surfaces that satisfy this assumption, the simulation results show that the proposed method produces quite accurate 3D field solutions.

© 2012 Optical Society of America

OCIS Codes
(090.0090) Holography : Holography
(090.1760) Holography : Computer holography
(090.1995) Holography : Digital holography
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Fourier Optics and Signal Processing

Original Manuscript: March 9, 2012
Manuscript Accepted: April 26, 2012
Published: June 29, 2012

Erdem Şahin and Levent Onural, "Scalar diffraction field calculation from curved surfaces via Gaussian beam decomposition," J. Opt. Soc. Am. A 29, 1459-1469 (2012)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. L. Onural, F. Yaraş, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011). [CrossRef]
  2. N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).
  3. J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16, 12372–12386 (2008). [CrossRef]
  4. W. J. Dallas, “Computer-generated holograms,” in Digital Holography and Three-Dimensional Display, T. C. Poon, ed. (Springer, 2006), pp. 1–49.
  5. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005). [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), 2nd ed.
  7. J. D. Gaskill, “The propagation and diffraction of optical wave fields,” in Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 10.
  8. J. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966). [CrossRef]
  9. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976). [CrossRef]
  10. K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000). [CrossRef]
  11. M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992). [CrossRef]
  12. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005). [CrossRef]
  13. M. Janda, I. Hanák, and L. Onural, “Hologram synthesis for photorealistic reconstruction,” J. Opt. Soc. Am. A 25, 3083–3096 (2008). [CrossRef]
  14. L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).
  15. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57, 546–547 (1967). [CrossRef]
  16. G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).
  17. G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011). [CrossRef]
  18. D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).
  19. M. J. Bastiaans, “Gabor’s signal expansion and the Zak transform,” Appl. Opt. 33, 5241–5255 (1994). [CrossRef]
  20. M. J. Bastiaans, “Expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).
  21. L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A 28, 290–295 (2011). [CrossRef]
  22. P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).
  23. M. J. Bastiaans, “Oversampling in Gabor’s signal expansion by an integer factor,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, 1994 (IEEE, 1994), pp. 280–283.
  24. A. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981). [CrossRef]
  25. K. Tang, Mathematical Methods for Engineers and Scientists: Fourier Analysis, Partial Differential Equations and Variational Models (Springer, 2007).
  26. M. J. Bastiaans and A. J. van Leest, “Gabor’s signal expansion and the Gabor transform based on a non-orthogonal sampling geometry,” in Sixth International Symposium on Signal Processing and Its Applications, 2001 (IEEE, 2001), pp. 162–163.

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