Paraxial and nonparaxial polynomial beams and the analytic approach to propagation

Mark R. Dennis, Joerg B. Goette, Robert P. King, Michael A. Morgan, Miguel A. Alonso

Research output: Contribution to journalArticle (Academic Journal)peer-review

12 Citations (Scopus)


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. (C) 2011 Optical Society of America

Original languageEnglish
Pages (from-to)4452-4454
Number of pages3
JournalOptics Letters
Issue number22
Publication statusPublished - 15 Nov 2011


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