Paraxial and nonparaxial polynomial beams and the analytic approach to propagation

摘要

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.