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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


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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

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