OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 23, Iss. 3 — Mar. 1, 2006
  • pp: 730–740

Expansions for irradiance distribution near the focus in systems of different Fresnel numbers

Yajun Li  »View Author Affiliations

JOSA A, Vol. 23, Issue 3, pp. 730-740 (2006)

View Full Text Article

Enhanced HTML    Acrobat PDF (1145 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



It is shown in an earlier paper dealing with flat-topped light beams [ Opt. Lett. 27, 1007 (2002) ] that the profile of flat-topped beams can be expressed in the form 1 [ 1 exp ( ξ 2 ) ] M , where ξ is a dimensionless parameter and M is a nonnegative number. The expansion of the proposed expression is a finite series containing only the lowest-order Gaussian modes. This situation provides the possibility of reformulating the scalar theory of diffraction at an aperture in an opaque screen if the Gaussian mode expansion is employed to describe the boundary values of the light incident on the screen. As an example of this effort, an asymptotic model is established for three-dimensional irradiance distributions near the focus in systems of different Fresnel numbers. The proposed expansions contain only elementary functions and permit all elementary operations; therefore no special functions or special algorithms are needed in the evaluation of either irradiance distributions or the integrated energy in a focused field.

© 2006 Optical Society of America

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(110.1220) Imaging systems : Apertures
(260.0260) Physical optics : Physical optics
(260.1960) Physical optics : Diffraction theory

ToC Category:
Physical Optics

Original Manuscript: June 29, 2005
Manuscript Accepted: August 10, 2005

Yajun Li, "Expansions for irradiance distribution near the focus in systems of different Fresnel numbers," J. Opt. Soc. Am. A 23, 730-740 (2006)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. F. Gori, 'Flattened Gaussian beams,' Opt. Commun. 107, 335-341 (1994). [CrossRef]
  2. Y. Li, 'Light beams with flat-topped profiles,' Opt. Lett. 27, 1007-1009 (2002). [CrossRef]
  3. Y. Li, 'New expressions for flat-topped light beams,' Opt. Commun. 206, 225-334 (2002). [CrossRef]
  4. Y. Li, 'Flat-topped light beams with non-circular cross-sections,' J. Mod. Opt. 50, 1957-1966 (2003). [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.2.
  6. G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U. Press, 1962), pp. 537-550.
  7. Y. Li and E. Wolf, 'Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,' J. Opt. Soc. Am. A 1, 801-808 (1984). [CrossRef]
  8. J. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155) on p. 4.
  9. Y. Li, 'Degeneracy in the Fraunhofer diffraction of truncated Gaussian beams,' J. Opt. Soc. Am. A 4, 1237-1244 (1987). [CrossRef]
  10. Y. Li and E. Wolf, 'Focal shift in focused truncated Gaussian beams,' Opt. Commun. 42, 151-156 (1982). [CrossRef]
  11. E. H. Linfoot and E. Wolf, 'Phase distribution near focus in an aberration-free diffraction image,' Proc. Phys. Soc. London, Sect. B 69, 823-832 (1956). [CrossRef]
  12. E. Wolf, 'Light distribution near focus in an error-free diffraction image,' Proc. R. Soc. London, Ser. A 204, 533-548 (1951). [CrossRef]
  13. Y. Li, 'Encircled energy of diffracted converging spherical waves,' J. Opt. Soc. Am. 73, 1101-1104 (1983). [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Sec. 4.5.1.
  15. F. W. J. Olver, Asymptotics and Special Functions (Academic, 1974), Chap. 1.
  16. Y. Li and F. T. S. Yu, 'Intensity distribution near the focus of an apertured focused Gaussian beam,' Opt. Commun. 70, 1-7 (1989). [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