# On beam models and their paraxial approximation

## TL;DR

This paper derives focused laser pulse solutions to the electromagnetic wave equation, compares paraxial and non-paraxial models, and highlights limitations of the paraxial approximation for Lorentzian transverse profiles, with implications for quantum electrodynamics.

## Contribution

It introduces a vector-potential method for more accurate laser beam modeling beyond the paraxial approximation, especially for Lorentzian transverse momentum distributions.

## Key findings

- Paraxial approximation works near the focal axis within a Rayleigh range.
- Paraxial models incorrectly predict transverse fall-off for Lorentzian profiles.
- The vector-potential approach improves modeling accuracy for quantum electrodynamics applications.

## Abstract

We derive focused laser pulse solutions to the electromagnetic wave equation in vacuum. After reproducing beam and pulse expressions for the well-known paraxial Gaussian and axicon cases, we apply the method to analyse a laser beam with Lorentzian transverse momentum distribution. Whilst a paraxial approach has some success close to the focal axis and within a Rayleigh range of the focal spot, we find that it incorrectly predicts the transverse fall-off typical of a Lorentzian. Our vector-potential approach is particularly relevant to calculation of quantum electrodynamical processes in weak laser pulse backgrounds.