OSA's Digital Library

Optics Letters

Optics Letters

| RAPID, SHORT PUBLICATIONS ON THE LATEST IN OPTICAL DISCOVERIES

  • Editor: Alan E. Willner
  • Vol. 36, Iss. 22 — Nov. 15, 2011
  • pp: 4452–4454

Paraxial and nonparaxial polynomial beams and the analytic approach to propagation

Mark R. Dennis, Jörg B. Götte, Robert P. King, Michael A. Morgan, and Miguel A. Alonso  »View Author Affiliations


Optics Letters, Vol. 36, Issue 22, pp. 4452-4454 (2011)
http://dx.doi.org/10.1364/OL.36.004452


View Full Text Article

Enhanced HTML    Acrobat PDF (102 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We construct solutions of the paraxial and Helmholtz equations that are polynomials in their spatial variables. These are derived explicitly by using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates, respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable z with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of paraxially and nonparaxially propagating beams, especially in regard to the divergence of nonparaxial analogs to familiar paraxial beams.

© 2011 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(050.1960) Diffraction and gratings : Diffraction theory
(350.5500) Other areas of optics : Propagation

ToC Category:
Diffraction and Gratings

History
Original Manuscript: August 26, 2011
Manuscript Accepted: October 3, 2011
Published: November 15, 2011

Citation
Mark R. Dennis, Jörg B. Götte, Robert P. King, Michael A. Morgan, and Miguel A. Alonso, "Paraxial and nonparaxial polynomial beams and the analytic approach to propagation," Opt. Lett. 36, 4452-4454 (2011)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-22-4452


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. F. Nye, J. Opt. Soc. Am. A 15, 1132 (1998). [CrossRef]
  2. M. V. Berry, J. Mod. Opt. 45, 1845 (1998). [CrossRef]
  3. M. V. Berry and M. R. Dennis, J. Phys. A 34, 8877 (2001). [CrossRef]
  4. M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, Nat. Phys. 6, 118 (2010). [CrossRef]
  5. Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, and J. Tollaksen, J. Phys. A 44, 365304 (2011). [CrossRef]
  6. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975). [CrossRef]
  7. A. Wünsche, J. Opt. Soc. Am. A 9, 765 (1992). [CrossRef]
  8. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003). [CrossRef] [PubMed]
  9. A. Torre, J. Opt. 13, 015701 (2011). [CrossRef]
  10. R. Borghi, F. Gori, G. Guattari, and M. Santarsiero, Opt. Lett. 36, 963 (2011). [CrossRef] [PubMed]
  11. National Institute of Standards and Technology, Digital Library of Mathematical Functions (NIST, 2010), http://dlmf.nist.gov/.
  12. A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973). [CrossRef]
  13. D. V. Widder, The Heat Equation (Academic, 1976).
  14. E. Grosswald, Bessel Polynomials (Springer1979).
  15. S. Roman, The Umbral Calculus (Dover, 2005).
  16. L. R. Bragg and J. W. Dettman, Rocky Mt. J. Math. 25, 887(1995). [CrossRef]
  17. Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998). [CrossRef]
  18. C. J. R. Sheppard, J. Opt. Soc. Am. A 18, 1579 (2001). [CrossRef]
  19. A. V. Novitsky and D. V. Novitsky, Opt. Lett. 34, 3430 (2009). [CrossRef] [PubMed]

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