Scalar representation of paraxial and nonparaxial laser beams

TLDR: In this article, a theory of non-paraxial beam propagation in two and three dimensions is developed by the use of the Mathieu and oblate spheroidal wave functions, respectively.

Abstract: The development of technology of small dimensions requires a different treatment of electromagnetic beams with transverse dimensions of the order of the wavelength. These are the nonparaxial beams either in two or three spatial dimensions. Based on the Helmholtz equation, a theory of nonparaxial beam propagation in two and three dimensions is developed by the use of the Mathieu and oblate spheroidal wave functions, respectively. Mathieu wave functions are the solutions of the Helmholtz equation in planar elliptic coordinates that is a special case of the prolate spheroidal geometry. So we may simply refer to the solutions, either in two or three dimensions, as spheroidal wave functions. Besides the order mode, the spheroidal wave functions are characterized by a parameter that will be referred to as the spheroidal parameter. Divergence of the beam is characterized by choosing the numeric value of this spheroidal parameter, having a perfect control on the nonparaxial properties of the beam under study. When the spheroidal parameter is above a given threshold, the well known paraxial Laguerre-Gauss and Hermite-Gauss beams are recovered, in their respective dimensions. In other words, the spheroidal wave functions represent a unified theory that can describe electromagnetic beams in the nonparaxial regime as well as in the paraxial one.