Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer

TL;DR

This paper provides a clear, self-contained overview of Gaussian, Hermite-Gaussian, and Laguerre-Gaussian laser beam modes, introducing a differential operator method for deriving higher-order modes from the fundamental Gaussian beam.

Contribution

It presents a novel, didactic approach using differential operators on the fundamental Gaussian mode to derive various beam modes, including Bessel beams, differing from traditional methods.

Findings

Unified framework for deriving beam modes

Simplified derivation of special modes like Bessel beams

Educational presentation of laser beam mode theory

Abstract

The paper aims at presenting a didactic and self-contained overview of Gauss-Hermite and Gauss-Laguerre laser beam modes. The usual textbook approach for deriving these modes is to solve the Helmoltz electromagnetic wave equation within the paraxial approximation. Here, a different technique is presented: Using the plane wave representation of the fundamental Gaussian mode as seed function, all higher-order beam modes are derived by acting with differential operators on this fundamental solution. Even special beam modes as the recently introduced Bessel beams are easily described within that framework.