OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 5 — Mar. 2, 2009
  • pp: 3690–3697

Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre-Gaussian beams

Colin JR Sheppard  »View Author Affiliations

Optics Express, Vol. 17, Issue 5, pp. 3690-3697 (2009)

View Full Text Article

Enhanced HTML    Acrobat PDF (1003 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The concept of the dual of a beam is discussed. The duals of Bessel-Gauss beams, elegant Hermite- or Laguerre-Gaussian beams and generalized Hermite- or Laguerre-Gauss beams are described. Duality is considered within the framework of hypergeometric beams in Cartesian and polar coordinates. The connection with the “modified Laguerre-Gauss” beam is discussed.

© 2009 Optical Society of America

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Physical Optics

Original Manuscript: August 13, 2008
Revised Manuscript: December 11, 2008
Manuscript Accepted: January 7, 2009
Published: February 24, 2009

Colin J.R. Sheppard, "Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre- Gaussian beams," Opt. Express 17, 3690-3697 (2009)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. A 60, 1168-1177 (1970). [CrossRef]
  2. A. W. Lohmann, "Ein neues Dualitatsprinzip in der Optik," Optik 11, 478-488 (1954).
  3. A. W. Lohmann, "Duality in optics," Optik 89, 93-97 (1992).
  4. F. Riesz and B. Szökefalvi-Nagy, Functional Analysis (Dover, New York, 1990).
  5. M. J. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991). [CrossRef]
  6. A. W. Lohmann and D. Mendlovic, "Self-Fourier objects and other self-transform objects," J. Opt. Soc. Am. A 9, 2009-2012 (1992). [CrossRef]
  7. C. J. R. Sheppard and S. Saghafi, "Flattened light beams," Opt. Comm. 132, 144-152 (1996). [CrossRef]
  8. C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves Opt. Acoustics 2, 105-112 (1978). [CrossRef]
  9. F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987). [CrossRef]
  10. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
  11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).
  12. H. Kogelnik and T. Li, "Laser beams and resonators," App. Opt. 5, 1550-1567 (1966). [CrossRef]
  13. A. Wünsche, "Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen," Ann. Phys. 25, 113-135 (1970). [CrossRef]
  14. A. Wünsche, "Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry," J. Opt. Soc. Am. A 6, 1320-1329 (1989). [CrossRef]
  15. C. J. R. Sheppard, "High aperture beams," Journal of the Optical Society of America A 18, 1579-1587 (2001). [CrossRef]
  16. A. E. Siegman, "Hermite-Gaussian functions of complex arguments as optical-beam eigenfunctions," J. Opt. Soc. Am. 63, 1093-1094 (1973). [CrossRef]
  17. R. Pratesi and L. Ronchi, "Generalized Gaussian beams in free space," J. Opt. Soc. Am. 67, 1274-1276 (1977). [CrossRef]
  18. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1994).
  19. S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Optics 45, 1999-2009 (1998). [CrossRef]
  20. M. Porras, R. Borghi, and M. Santarsiero, "Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 177-184 (2001). [CrossRef]
  21. M. A. Bandres and J. C. Gutierrez-Vega, "Circular beams," Opt. Lett. 33, 177-179 (2008). [CrossRef] [PubMed]
  22. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007). [CrossRef] [PubMed]
  23. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian beams," Opt. Lett. 32, 3053-3055 (2007). [CrossRef] [PubMed]
  24. M. A. Bandres and J. C. Gutierrez-Vega, "Cartesian beams," Opt. Lett. 32, 3459-3461 (2007).25. [CrossRef] [PubMed]
  25. J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis," Opt. Lett. 11, 1521-1523 (2007). [CrossRef]
  26. J. C. Gutiérrez-Vega, "Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes," Optics Express 15, 6300-6313 (2007). [CrossRef] [PubMed]
  27. C. F. R. Caron and R. M. Potvliege, "Bessel-modulated Gaussian beams with quadratic radial dependence," Opt. Comm. 164, 83-93 (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.


Fig. 1. Fig. 2. Fig. 3.
Fig. 4.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited