Constrained Bateman-Hillion Solutions for Hermite-Gaussian Beams

TL;DR

This paper develops exact solutions for Hermite-Gaussian beams using Bateman-Hillion methods with a space-time constraint, revealing their evolution into angular functions on spherical phase fronts and linking wave and Schrödinger equations.

Contribution

It introduces a novel application of Bateman-Hillion solutions with a space-time constraint to Hermite-Gaussian beams, connecting wave and quantum equations under specific conditions.

Findings

Hermite-Gaussian functions become pure angular functions on spherical phase fronts

Wave and Schrödinger equations are interchangeable within the constrained space

Field density falls as inverse square of distance from the focal point

Abstract

Exact Bateman-Hillion solutions of the wave equation are applied to Hermite-Gaussian beams using a space-time constraint condition that requires the field density to fall as the inverse square of distance from the focal point of the beam at large distances from it. Following a familiar practice, the constraint is implemented in integrals through the use of a Dirac delta function. It is shown the Hermite-Gaussian functions evolve to become pure functions of angular position on the fully developed spherical phase fronts. Under the paraxial approximation it is further shown the wave equation and Schrodinger equation are interchangeable within the constraint space in correspondence to a recent paper claiming indirect evidence of Gouy phase in matter waves.