English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We construct solutions of the paraxial and Helmholtz equations which arepolynomials in their spatial variables. These are derived explicitly using theangular spectrum method and generating functions. Paraxial polynomials have theform of homogeneous Hermite and Laguerre polynomials in Cartesian andcylindrical coordinates respectively, analogous to heat polynomials for thediffusion equation. Nonparaxial polynomials are found by substituting monomialsin the propagation variable $z$ with reverse Bessel polynomials. These explicitanalytic forms give insight into the mathematical structure of paraxially andnonparaxially propagating beams, especially in regards to the divergence ofnonparaxial analogs to familiar paraxial beams.

Paraxial and nonparaxial polynomial beams and the analytic approach to propagation