Polynomials of Gaussians and vortex-Gaussian beams as complete, transversely confined bases

TL;DR

This paper introduces a new discrete basis for paraxial beams using monomial vortices multiplied by Gaussian polynomials, which maintains a consistent effective size regardless of element order, unlike traditional bases.

Contribution

The authors propose a novel basis combining monomial vortices and Gaussian polynomials that remains transversely confined and size-independent of the element order.

Findings

Basis elements have size roughly independent of order

Contrasts with Hermite-Gauss and Laguerre-Gauss modes

Facilitates efficient localized field expansion

Abstract

A novel type of discrete basis for paraxial beams is proposed, consisting of monomial vortices times polynomials of Gaussians in the radial variable. These bases have the distinctive property that the effective size of their elements is roughly independent of element order, meaning that the optimal scaling for expanding a localized field does not depend significantly on truncation order. This behavior contrasts with that of bases composed of polynomials times Gaussians, such as Hermite-Gauss and Laguerre-Gauss modes, where the scaling changes roughly as the inverse square root of the truncation order.