# The Vacuum Emission Picture Beyond Paraxial Approximation

## TL;DR

This paper develops a numerical method to study quantum vacuum nonlinearities caused by strong electromagnetic fields beyond the paraxial approximation, enabling analysis of generic laser pulse configurations in high-intensity laser experiments.

## Contribution

It introduces an efficient numerical solver that extends vacuum nonlinear optics studies beyond paraxial approximation using the locally constant field approximation.

## Key findings

- Allows calculation of signal photon emission in non-paraxial fields
- Enables analysis of realistic laser pulse configurations
- Provides a versatile tool for high-intensity laser experiments

## Abstract

Optical signatures of the effective nonlinear couplings among electromagnetic fields in the quantum vacuum can be conveniently described in terms of stimulated photon emission processes induced by strong classical, space-time dependent electromagnetic fields. Recent studies have adopted this approach to study collisions of Gaussian laser pulses in paraxial approximation. The present study extends these investigations beyond the paraxial approximation by using an efficient numerical solver for the classical input fields. This new numerical code allows for a consistent theoretical description of optical signatures of QED vacuum nonlinearities in generic electromagnetic fields governed by Maxwell's equations in the vacuum, such as manifestly non-paraxial laser pulses. Our code is based on a locally constant field approximation of the Heisenberg-Euler effective Lagrangian. As this approximation is applicable for essentially all optical high-intensity laser experiments, our code is capable of calculating signal photon emission amplitudes in completely generic input field configurations, limited only by numerical cost.