Exact solution of Helmholtz equation for the case of non-paraxial Gaussian beams

TL;DR

This paper presents an exact solution to the 3D Helmholtz equation for non-paraxial Gaussian beams using spherical coordinates, revealing conditions for optical vortices and advancing understanding of complex wave structures.

Contribution

It introduces a novel exact solution for non-paraxial Gaussian beams in 3D Helmholtz equation, linking amplitude zeros to optical vortices.

Findings

Exact solution satisfies Riccati equations for Gaussian beams

Amplitude zero indicates optical vortex presence

Solution enhances understanding of non-paraxial beam structures

Abstract

A new type of exact solutions of the full 3 dimensional spatial Helmholtz equation for the case of non-paraxial Gaussian beams is presented here. We consider appropriate representation of the solution for Gaussian beams in a spherical coordinate system by substituting it to the full 3 dimensional spatial Helmholtz Equation. Analyzing the structure of the final equation, we obtain that governing equations for the components of our solution are represented by the proper Riccati equations of complex value, which has no analytical solution in general case. But we find one of the possible exact solution which is proved to satisfy to such an equations for Gaussian beams. Decreasing of the amplitude A of presented solution up to the zero (at the appropriate meaning of angle parameter) could be associated with the existence of an optical vortex at this point. Optical vortex (also known as a "dislocation in wave trains") is a zero of an optical field, a point of zero intensity.