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

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.