Theory of electromagnetic beams

TLDR

The paper assumes a plane‑polarized vector potential whose nonvanishing component satisfies a scalar wave equation. The study presents a simple method for calculating paraxial electromagnetic beam properties and derives explicit second‑ and third‑order field corrections. The method expands the vector potential in powers of \(w_{0}/l\) to obtain a formal solution for the beam fields. The approach reproduces the lowest‑order Gaussian beam fields and yields explicit second‑ and third‑order corrections to the transverse and longitudinal electric fields.

Abstract

A relatively simple method for calculating the properties of a paraxial beam of electromagnetic radiation propagating in vacuum is presented. The central idea of the paper is that the vector potential field is assumed to be plane-polarized. The nonvanishing component of the vector potential obeys a scalar wave equation. A formal solution employing an expansion in powers of $\frac{{w}_{0}}{l}$ is obtained, where ${w}_{0}$ is the beam waist and $l$ the diffraction length. This gives the same result for the lowest-order components of the transverse and longitudinal electric field of a Gaussian beam that was derived by Lax, Louisell, and McKnight using a more complicated approach. We derive explicit expressions for the second-order transverse electric field and the third-order longitudinal field corrections.