A concise and universal method for deriving arbitrary paraxial and non-paraxial cylindrical Gaussian-type light modes

TL;DR

The paper introduces a universal, concise method based on the Hankel transform for deriving a wide range of cylindrical Gaussian-type light modes, including new beam types, applicable to both paraxial and non-paraxial regimes.

Contribution

It presents a novel, unified approach for deriving various cylindrical Gaussian-like beams using the Hankel transform, enabling easy generation of known and new beam solutions.

Findings

Derived expressions for multiple Gaussian-like beams.

Introduced a new $\gamma$ beam based on incomplete gamma functions.

Demonstrated the method's applicability to both paraxial and non-paraxial equations.

Abstract

A concise method of deriving expressions for Gaussian-like solutions of the paraxial and d'Alembert equations is presented. This method is based on the Hankel transform. Choosing some Gaussian base functions with slight modifications of the prefactor all basic beams of cylindrical character can be easily obtained. This refers to Gaussian, Bessel-Gaussian, modified Bessel-Gaussian, Laguerre-Gaussian and Kummer-Gaussian (i.e., Hypergeometric-Gaussian) beams although potentially other beams can come into play as well. For instance a new type of a beam that can be derived in this way is described through the incomplete gamma function so it may be called a $\gamma$ beam.