OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 6 — Jun. 1, 2013
  • pp: 1180–1183

Rayleigh–Sommerfeld diffraction formula in k space

C. J. R. Sheppard, J. Lin, and S. S. Kou  »View Author Affiliations

JOSA A, Vol. 30, Issue 6, pp. 1180-1183 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (110 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



An angular spectrum representation in three dimensions is used to develop three-dimensional Fourier forms of the first and second Rayleigh–Sommerfeld diffraction formulae and the Kirchhoff diffraction formula. For forward-propagating waves, these reduce to three-dimensional Fourier representations for diffraction in the forward half-space.

© 2013 Optical Society of America

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(260.1960) Physical optics : Diffraction theory
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Diffraction and Gratings

Original Manuscript: March 12, 2013
Revised Manuscript: April 28, 2013
Manuscript Accepted: April 28, 2013
Published: May 16, 2013

C. J. R. Sheppard, J. Lin, and S. S. Kou, "Rayleigh–Sommerfeld diffraction formula in k space," J. Opt. Soc. Am. A 30, 1180-1183 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. H. Weyl, “Ausbreitung elektromagnetische Wellen über einem ebenen Leiter,” Ann. Phys. 365, 481–500 (1919). [CrossRef]
  2. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964). [CrossRef]
  3. C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).
  4. P. Andrés, M. Martinez-Corral, and J. Ojeda-Castañeda, “Off-axis focal shift for rotationally nonsymmetric screens,” Opt. Lett. 18, 1290–1293 (1993). [CrossRef]
  5. J. Lin, X. C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffraction field,” Opt. Lett. 36, 1341–1343 (2011). [CrossRef]
  6. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969). [CrossRef]
  7. W. H. Carter, “Reconstruction of scattering objects from holograms,” J. Opt. Soc. Am. 60, 306–314 (1970). [CrossRef]
  8. H. Arnoldus, “Representation of the near-field, middle-field, and far-field electromagnetic Green’s functions in reciprocal space,” J. Opt. Soc. Am. B 18, 547–555 (2001). [CrossRef]
  9. R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill Electrical and Electronic Engineering Series (McGraw-Hill, 1978).
  10. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1978).
  11. T. Tyc and X. Zhang, “Perfect lenses in focus,” Nature 480, 42–43 (2011). [CrossRef]
  12. J. D. Jackson, Classical Electrodynamics (Wiley, 1962).
  13. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University, 1981).
  14. A. Sommerfeld, “Die Greensche Funktion der Schwingungsgleichung,” Jahresbericht der Deutschen Mathematiker-Vereinigung 21, 309–353 (1912).
  15. A. Sommerfeld, Optics, Lectures on Theoretical Physics (Academic, 1964).
  16. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999).
  17. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  19. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57, 546–547 (1967). [CrossRef]
  20. C. L. Andrews, “Diffraction pattern of a circular aperture at short distances,” Phys. Rev. 71, 777–786 (1947). [CrossRef]
  21. 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]
  22. S. S. Kou, C. J. R. Sheppard, and J. Lin, “Exact evaluation of a volumetric diffracted field with 3D convolution: the 3D angular spectrum method,” Opt. Lett. (submitted).

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