OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 27, Iss. 2 — Feb. 1, 2010
  • pp: 327–332

Normalization of optical Weber waves and Weber–Gauss beams

B. M. Rodríguez-Lara  »View Author Affiliations

JOSA A, Vol. 27, Issue 2, pp. 327-332 (2010)

View Full Text Article

Enhanced HTML    Acrobat PDF (338 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The normalization of energy divergent Weber waves and finite energy Weber–Gauss beams is reported. The well-known Bessel and Mathieu waves are used to derive the integral relations between circular, elliptic, and parabolic waves and to present the Bessel and Mathieu wave decomposition of the Weber waves. The efficiency to approximate a Weber–Gauss beam as a finite superposition of Bessel–Gauss beams is also given.

© 2010 Optical Society of America

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(140.3300) Lasers and laser optics : Laser beam shaping
(260.1960) Physical optics : Diffraction theory
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Physical Optics

Original Manuscript: November 12, 2009
Revised Manuscript: December 11, 2009
Manuscript Accepted: December 13, 2009
Published: January 29, 2010

B. M. Rodríguez-Lara, "Normalization of optical Weber waves and Weber-Gauss beams," J. Opt. Soc. Am. A 27, 327-332 (2010)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, 1927).
  2. J. A. Stratton, Electromagnetic Theory, International Series in Pure and Applied Physics (Read Books, 1941).
  3. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1, International Series in Pure and Applied Physics (McGraw-Hill, 1953).
  4. W. Miller, Symmetry and Separation of Variables, Encyclopedia of Mathematics and Its Applications (Cambridge Univ. Press, 1984).
  5. J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987). [CrossRef]
  6. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
  7. A. A. Makarov, J. A. Smorodinsky, K. Valiev, and P. Winternitz, “A systematic search for nonrelativistic systems with dynamical symmetries,” Il Nuovo Cimento LII A, 1061-1084 (1967).
  8. C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35-80 (1976).
  9. R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71, 033411 (2005). [CrossRef]
  10. K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8, 867-877 (2006). [CrossRef]
  11. B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78, 033813 (2008). [CrossRef]
  12. B. M. Rodríguez-Lara and R. Jáuregui, “Dynamical constants of structured photons with parabolic-cylindrical symmetry,” Phys. Rev. A 79, 055806 (2009). [CrossRef]
  13. D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2009). [PubMed]
  14. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289-298 (2005). [CrossRef]
  15. C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 456, 068001 (2006). [CrossRef]
  16. M. B. Alvarez-Elizondo, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, “Generation of Mathieu-Gauss modes with an axicon-based laser resonator,” Opt. Express 16, 18770-18775 (2008). [CrossRef]
  17. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44-46 (2004). [CrossRef] [PubMed]
  18. J. C. Gutiérrez-Vega and M. A. Bandres, “Normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A 24, 215-220 (2007). [CrossRef]
  19. A. P. Prudnikov, J. A. Brychkov, and O. I. Marichev, Integrals and Series, Vols. 1-3 (Mockba, 1981).
  20. A.Erdélyi, ed., Higher Trascendental Functions, Vol. 1 (McGraw-Hill, 1985).
  21. A. A. Inayat-Hussain, “Mathieu integral transforms,” J. Math. Phys. 32, 669-675 (1991). [CrossRef]
  22. N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Courier Dover, 1970).

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

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited