Paraxial and nonparaxial polynomial beams and the analytic approach to propagation

TL;DR

This paper develops explicit polynomial solutions for paraxial and nonparaxial wave equations using analytic methods, providing insights into beam propagation and divergence in optical physics.

Contribution

It introduces a systematic construction of polynomial solutions for both paraxial and Helmholtz equations, linking them to Hermite, Laguerre, and Bessel polynomials.

Findings

Explicit forms of paraxial polynomials as Hermite and Laguerre polynomials

Nonparaxial polynomials derived via reverse Bessel polynomials

Enhanced understanding of beam divergence and propagation structure

Abstract

We construct solutions of the paraxial and Helmholtz equations which are polynomials in their spatial variables. These are derived explicitly 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 regards to the divergence of nonparaxial analogs to familiar paraxial beams.